Mathematics and Philosophy - dl.booktolearn.comdl.booktolearn.com/ebooks2/science/philosophy/... · xii Mathematics and Philosophy parentheses and embarking on a quest for a hypothetical

Mathematics and Philosophy

Series Editor Nikolaos Limnios

Mathematics and Philosophy

Daniel Parrochia

First published 2018 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

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Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Part 1. The Contribution of Mathematician–Philosophers . . . . . . . 1

Introduction to Part 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Chapter 1. Irrational Quantities . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1. The appearance of irrationals or the end of the Pythagorean dream . . 81.2. The first philosophical impact . . . . . . . . . . . . . . . . . . . . . . . 91.3. Consequences of the discovery of irrationals . . . . . . . . . . . . . . . 11

1.3.1. The end of the eternal return . . . . . . . . . . . . . . . . . . . . . . 111.3.2. Abandoning the golden ratio . . . . . . . . . . . . . . . . . . . . . . 111.3.3. The problem of disorder in medicine, morals and politics . . . . . . 12

1.4. Possible solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.5. A famous example: the golden number . . . . . . . . . . . . . . . . . . 141.6. Plato and the dichotomic processes . . . . . . . . . . . . . . . . . . . . 161.7. The Platonic generalization of ancient Pythagoreanism . . . . . . . . . 17

1.7.1. The Divided Line analogy . . . . . . . . . . . . . . . . . . . . . . . 171.7.2. The algebraic interpretation . . . . . . . . . . . . . . . . . . . . . . 18

1.7.2.1. Impossibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.7.2.2. The case where k = ! . . . . . . . . . . . . . . . . . . . . . . . 19

1.8. Epistemological consequences: the evolution of reason . . . . . . . . . 20

Chapter 2. All About the Doubling of the Cube . . . . . . . . . . . . . . 23

2.1. History of the question of doubling a cube . . . . . . . . . . . . . . . . 242.2. The non-rationality of the solution . . . . . . . . . . . . . . . . . . . . . 24

2.2.1. Demonstration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2.2. The diagonal is not a solution . . . . . . . . . . . . . . . . . . . . . 25

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2.3. The theory proposed by Hippocrates of Chios . . . . . . . . . . . . . . 252.4. A philosophical application: platonic cosmology . . . . . . . . . . . . 272.5. The problem and its solutions . . . . . . . . . . . . . . . . . . . . . . . 29

2.5.1. The future of the problem . . . . . . . . . . . . . . . . . . . . . . . . 292.5.2. Some solutions proposed by authors of the classical age . . . . . . 30

2.5.2.1. Mechanical solutions . . . . . . . . . . . . . . . . . . . . . . . 302.5.2.2. Analytical solution . . . . . . . . . . . . . . . . . . . . . . . . 31

2.5.3. The doubling of the cube – going beyond Archytas: the evolutionof mathematical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.5.3.1. Menaechmus’ solution . . . . . . . . . . . . . . . . . . . . . . 372.5.3.2. A brief overview of the other solutions . . . . . . . . . . . . . 39

2.6. The trisection of an angle . . . . . . . . . . . . . . . . . . . . . . . . . . 402.6.1. Bold mathematicians . . . . . . . . . . . . . . . . . . . . . . . . . . 402.6.2. Plato, the tripartition of the soul and self-propulsion . . . . . . . . . 422.6.3. A very essential shell . . . . . . . . . . . . . . . . . . . . . . . . . . 442.6.4. A final excercus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.7. Impossible problems and badly formulated problems . . . . . . . . . . 462.8. The modern demonstration . . . . . . . . . . . . . . . . . . . . . . . . . 47

Chapter 3. Quadratures, Trigonometry and Transcendance . . . . . 51

3.1. " – the mysterious number . . . . . . . . . . . . . . . . . . . . . . . . . 523.2. The error of the “squarers” . . . . . . . . . . . . . . . . . . . . . . . . . 533.3. The explicit computation of " . . . . . . . . . . . . . . . . . . . . . . . 553.4. Trigonometric considerations . . . . . . . . . . . . . . . . . . . . . . . . 573.5. The paradoxical philosophy of Nicholas of Cusa . . . . . . . . . . . . . 59

3.5.1. An attempt at computing an approximate value for " . . . . . . . . 593.5.2. Philosophical extension . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.6. What came next and the conclusion to the history of " . . . . . . . . . 633.6.1. The age of infinite products . . . . . . . . . . . . . . . . . . . . . . . 643.6.2. Machin’s algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.6.3. The problem of the nature of " . . . . . . . . . . . . . . . . . . . . . 653.6.4. Numerical and philosophical transcendance: Kant, Lambert andLegendre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Part 2. Mathematics Becomes More Powerful . . . . . . . . . . . . . . . 69


Chapter 4. Exploring Mathesis in the 17th Century . . . . . . . . . . . 75

4.1. The innovations of Cartesian mathematics . . . . . . . . . . . . . . . . 764.2. The “plan” for Descartes’ Geometry . . . . . . . . . . . . . . . . . . . . 794.3. Studying the classification of curves . . . . . . . . . . . . . . . . . . . . 79

4.3.1. Possible explanations for the mistakes made by the Ancients . . . . 81

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4.3.2. Conditions for the admissibility of curves in geometry . . . . . . . 834.4. Legitimate constructions . . . . . . . . . . . . . . . . . . . . . . . . . . 854.5. Scientific consequences of Cartesian definitions . . . . . . . . . . . . . 874.6. Metaphysical consequences of Cartesian mathematics . . . . . . . . . . 88

Chapter 5. The Question of Infinitesimals . . . . . . . . . . . . . . . . . 91

5.1. Antiquity – the prehistory of the infinite . . . . . . . . . . . . . . . . . 925.1.1. Infinity as Anaximander saw it . . . . . . . . . . . . . . . . . . . . . 925.1.2. The problem of irrationals and Zeno’s paradoxes . . . . . . . . . . 935.1.3. Aristotle and the dual nature of the Infinite . . . . . . . . . . . . . . 96

5.2. The birth of the infinitesimal calculus . . . . . . . . . . . . . . . . . . . 985.2.1. Newton’s Writings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.2.2. Leibniz’s contribution . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.2.3. The impact of calculus on Leibnizian philosophy . . . . . . . . . . 105

5.2.3.1. Small perceptions and differentials . . . . . . . . . . . . . . . 1055.2.3.2. Matter and living beings . . . . . . . . . . . . . . . . . . . . . 1095.2.3.3. The image of order . . . . . . . . . . . . . . . . . . . . . . . . 110

5.2.4. The epistemological problem . . . . . . . . . . . . . . . . . . . . . . 117

Chapter 6. Complexes, Logarithms and Exponentials . . . . . . . . . 121

6.1. The road to complex numbers . . . . . . . . . . . . . . . . . . . . . . . 1226.2. Logarithms and exponentials . . . . . . . . . . . . . . . . . . . . . . . . 1256.3. De Moivre’s and Euler’s formulas . . . . . . . . . . . . . . . . . . . . . 1286.4. Consequences on Hegelian philosophy . . . . . . . . . . . . . . . . . . 1306.5. Euler’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1326.6. Euler, Diderot and the existence of God . . . . . . . . . . . . . . . . . . 1336.7. The approximation of functions . . . . . . . . . . . . . . . . . . . . . . 134

6.7.1. Taylor’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1356.7.2. MacLaurin’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.8. Wronski’s philosophy and mathematics . . . . . . . . . . . . . . . . . . 1376.8.1. The Supreme Law of Mathematics . . . . . . . . . . . . . . . . . . 1386.8.2. Philosophical interpretation . . . . . . . . . . . . . . . . . . . . . . 142

6.9. Historical positivism and spiritual metaphysics . . . . . . . . . . . . . . 1436.9.1. Comte’s vision of mathematics . . . . . . . . . . . . . . . . . . . . . 1436.9.2. Renouvier’s reaction . . . . . . . . . . . . . . . . . . . . . . . . . . 1466.9.3. Spiritualist derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.10. The physical interest of complex numbers . . . . . . . . . . . . . . . . 1486.11. Consequences on Bergsonian philosophy . . . . . . . . . . . . . . . . 150

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Part 3. Significant Advances . . . . . . . . . . . . . . . . . . . . . . . . . . 155


Chapter 7. Chance, Probability and Metaphysics . . . . . . . . . . . . 161

7.1. Calculating probability: a brief history . . . . . . . . . . . . . . . . . . 1627.2. Pascal’s “wager” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

7.2.1. The Pensées passage . . . . . . . . . . . . . . . . . . . . . . . . . . 1667.2.2. The formal translation . . . . . . . . . . . . . . . . . . . . . . . . . . 1677.2.3. Criticism and commentary . . . . . . . . . . . . . . . . . . . . . . . 167

7.2.3.1. Laplace’s criticism . . . . . . . . . . . . . . . . . . . . . . . . . 1677.2.3.2. Emile Borel’s observation . . . . . . . . . . . . . . . . . . . . 1697.2.3.3. Decision theory . . . . . . . . . . . . . . . . . . . . . . . . . . 1707.2.3.4. The non-standard analysis framework . . . . . . . . . . . . . . 171

7.3. Social applications, from Condorcet to Musil . . . . . . . . . . . . . . . 1727.4. Chance, coincidences and omniscience . . . . . . . . . . . . . . . . . . 174

Chapter 8. The Geometric Revolution . . . . . . . . . . . . . . . . . . . . 179

8.1. The limits of the Euclidean demonstrative ideal . . . . . . . . . . . . . 1808.2. Contesting Euclidean geometry . . . . . . . . . . . . . . . . . . . . . . 1838.3. Bolyai’s and Lobatchevsky geometries . . . . . . . . . . . . . . . . . . 1848.4. Riemann’s elliptical geometry . . . . . . . . . . . . . . . . . . . . . . . 1918.5. Bachelard and the philosophy of “non” . . . . . . . . . . . . . . . . . . 1948.6. The unification of Geometry by Beltrami and Klein . . . . . . . . . . . 1968.7. Hilbert’s axiomatization . . . . . . . . . . . . . . . . . . . . . . . . . . . 1988.8. The reception of non-Euclidean geometries . . . . . . . . . . . . . . . . 2008.9. A distant impact: Finsler’s philosophy . . . . . . . . . . . . . . . . . . 200

Chapter 9. Fundamental Sets and Structures . . . . . . . . . . . . . . . 203

9.1. Controversies surrounding the infinitely large . . . . . . . . . . . . . . 2039.2. The concept of “the power of a set” . . . . . . . . . . . . . . . . . . . . 207

9.2.1. The “countable” and the “continuous” . . . . . . . . . . . . . . . . 2089.2.2. The uniqueness of the continuum . . . . . . . . . . . . . . . . . . . 2099.2.3. Continuum hypothesis and generalized continuum hypothesis . . . 212

9.3. The development of set theory . . . . . . . . . . . . . . . . . . . . . . . 2139.4. The epistemological route and others . . . . . . . . . . . . . . . . . . . 2189.5. Analytical philosophy and its masters . . . . . . . . . . . . . . . . . . . 2229.6. Husserl with Gödel? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2259.7. Appendix: Gödel’s ontological proof . . . . . . . . . . . . . . . . . . . 226

Contents ix

Part 4. The Advent of Mathematician-Philosophers . . . . . . . . . . . 229


Chapter 10. The Rise of Algebra . . . . . . . . . . . . . . . . . . . . . . . 233

10.1. Boolean algebra and its consequences . . . . . . . . . . . . . . . . . . 23410.2. The birth of general algebra . . . . . . . . . . . . . . . . . . . . . . . . 23710.3. Group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23810.4. Linear algebra and non-commutative algebra . . . . . . . . . . . . . . 24110.5. Clifford: a philosopher-mathematician . . . . . . . . . . . . . . . . . . 245

Chapter 11. Topology and Differential Geometry . . . . . . . . . . . . . 253

11.1. Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25311.1.1. Continuity and neighborhood . . . . . . . . . . . . . . . . . . . . . 25411.1.2. Fundamental definitions and theorems . . . . . . . . . . . . . . . . 25511.1.3. Properties of topological spaces . . . . . . . . . . . . . . . . . . . 25711.1.4. Philosophy of classifications versus topology of the being . . . . . 261

11.2. Models of differential geometry . . . . . . . . . . . . . . . . . . . . . 26211.2.1. Space as a support to thought . . . . . . . . . . . . . . . . . . . . . 26211.2.2. The general concept of manifold . . . . . . . . . . . . . . . . . . . 26311.2.3. The formal concept of differential manifold . . . . . . . . . . . . . 26411.2.4. The general theory of differential manifold . . . . . . . . . . . . . 26511.2.5. G-structures and connections . . . . . . . . . . . . . . . . . . . . . 266

11.3. Some philosophical consequences . . . . . . . . . . . . . . . . . . . . 26811.3.1. Whitehead’s philosophy and relativity . . . . . . . . . . . . . . . . 26911.3.2. Lautman’s singular work . . . . . . . . . . . . . . . . . . . . . . . 27011.3.3. Thom and the catastrophe theory . . . . . . . . . . . . . . . . . . . 273

Chapter 12. Mathematical Research and Philosophy . . . . . . . . . 279

12.1. The different domains . . . . . . . . . . . . . . . . . . . . . . . . . . . 27912.2. The development of classical mathematics . . . . . . . . . . . . . . . 28212.3. Number theory and algebra . . . . . . . . . . . . . . . . . . . . . . . . 28212.4. Geometry and algebraic topology . . . . . . . . . . . . . . . . . . . . 28412.5. Category and sheaves: tools that help in globalization . . . . . . . . . 286

12.5.1. Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28612.5.2. The Sheaf theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29212.5.3. Link to philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . 29412.5.4. Philosophical impact . . . . . . . . . . . . . . . . . . . . . . . . . . 295

12.6. Grothendieck’s unitary vision . . . . . . . . . . . . . . . . . . . . . . . 29512.6.1. Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29512.6.2. Topoi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29612.6.3. Motives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29812.6.4. Philosophical consequences of motives . . . . . . . . . . . . . . . 301

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Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

Introduction

Philosophy is not descended from heaven. It does not follow a completelyautonomous line of thought or a mode of speculation that is unknown to this world.Experience has shown us that the problems, concepts and theories of philosophy areborn out of a certain economic and political context, in close conjunction withsources of knowledge that fall within positive learning and practices. It is withinthese sites that philosophy normally discovers the inductive elements for its thinking.This is where, as they say, it finds life. A little historical context, therefore, oftenmakes it possible to reconstitute these elements that may sometimes leap off thesurface of a text but always inform its internal working. All we have to do is identifythem. Thus, metaphysics, from Plato to Husserl and beyond, has largely benefitedfrom advances made in an essential field of knowledge: mathematics. Any progressand revolution in this discipline has always provided philosophy with not onlyschools of thought, but also tools and instruments of thinking.

This is why we will study here the link between philosophy and the disciplineof mathematics, which is today an immense reservoir of extremely refined structureswith multiple interconnections. We will examine the vicissitudes of this relationshipthrough history. But the central question will be that of the knowledge that today canbe drawn from this discipline, which has lately become so powerful and complexthat it often and in large part soars out of reach of the knowledge and understandingof the philosopher. How can contemporary mathematics serve today’s philosophy?This is the real question that this book explores, being neither entirely an history ofphilosophy, nor an history of the sciences, and even less so that of epistemology.

We will not study science, its methods and laws, its evaluation or its status in thefield of knowledge. We will simply ask how this science can still be of use tophilosophers today in building a new vision of the world, and what this might be. Areader who is a philosopher will, therefore, certainly be asked to invert their thinkingand reject their usual methods. Rather than placing scientific knowledge entre

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parentheses and embarking on a quest for a hypothetical other knowledge, assumedto be more remarkable, more native or more radical (the method called the“phenomenological method”), we prefer suspending judgment, using the epoché(reduction) method for phenomenology itself and sticking to the only effectiveknowledge that truly makes up reason (or, at any rate, a considerable part of reason):mathematical knowledge. This knowledge contains within itself the most remarkabledevelopments and transformations not only of thought, but also of the world. Thisknowledge, by itself, has the capacity of constructing, in a methodical and reflectivemanner, the basic conceptual architecture needed to create worldviews. It wouldseem that philosophers have long forgotten this elementary humility that consists ofbeginning only with which is proven, instead of developing, through a blindadherence to empiricism, theories and dogma that lasted only a season, failing thetest of time, their weaknesses revealed over the course of history.

In doing this, we follow in the footsteps of thinkers who are more or less forgottentoday, but who kept repeating exactly what we say here. Gaston Milhaud, for example,had already noted this remarkable influence. In the opening lesson of a course taughtat Montpellier in 1908–1909, which was then published in the Revue Philosophiqueand reprinted in one of his books [MIL 11, pp. 21–22], we find the following text:

“My intention is to bind myself to certain essential characteristics ofmathematical thought and, above all, to study the repercussions it hashad on the concepts and doctrines of philosophers and even on the mostgeneral tendencies of the human mind.

How can we doubt that these repercussions have been significant whenhistory shows us mathematical speculations and philosophical reflectionsoften united in the same mind; when so often, from the Pythagoreans tothinkers such as Descartes, Leibniz, Kant and Renouvier (to speak onlyof the dead), some fundamental doctrines, at least, have been based on theidea of mathematics; when on all sides and in all times we see the seedsof not only critical views, but even systems that weigh in on the mostdifficult and obscure metaphysical problems and which reveal especially,through the justifications offered by the authors, a sort of vertigo born outof the manipulation of or just coming into contact with the speculationsof geometricians? The excitation in a thinker’s mind, far from being anaccident in the history of ideas, appears to us as a continuous and almostuniversal fact”.

A few years later, in 1912, Léon Brunschvicg published Les étapes de laphilosophie mathématique (Stages in Mathematical Philosophy), a book in which, asJean-Toussaint Desanti noted in his preface to the 1981 reprint, it clearly appears that

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mathematics informed philosophy1. In this book, hailed by Borel as “one of the mostpowerful attempts by any philosopher to assimilate a discipline as vast asmathematical science”, we can already see, as Desanti recalls, that “the slowemergence of forms of mathematical intelligibility provided the reader with a gridthrough which to interpret the history of different philosophies”. [BRU 81, p. VII].The fact remains, of course, that these two effects were secondary to Brunschvicg’schief project: to give an account of mathematical discourse itself in its operationalkernels, where the forms of construction of intelligible objects take place and wherethe activity of judgment (which he found so important) chiefly manifests itself, alongwith the dynamism inherent to the human intellect.

Admittedly, today mathematics is no longer accepted as truth in itself. Shaken toits foundations and now seen as being multivarious, if not uncertain2, it has seen itsrelevance diminish further of late. Knowing that 95% of truths are not demonstrablewithin our current systems and that the more complex a formula the more random itis3, we may well wonder as to the philosophical interest of the discipline. And so,Brunschvig’s concluding remark, according to which, “the free and fertile work ofthought dates back to the time when mathematics gave man the true norm for thetruth” [BRU 81, p. 577], may well make us smile. His Spinozian inspiration seemsquite passé now and the lazy philosopher will delight in stepping into the breach.

Nonetheless, not even recent masters – Jules Vuillemin, Gilles-Gaston Granger,Roshdi Rashed – who dedicated a large part of their work to mathematical thoughtand its philosophical consequences, have gone down this path. If they are often close,it is in the sense that their work generally looks at measuring the influence, or eventruly the impact, of mathematics on philosophy4. We will thus content ourselves withmodestly following in their path. This book will thus undoubtedly follow a countercurrent. However, it joins certain observations made by contemporarymathematicians in the wake of Bachelard. “The truth is that science enriches andrenews philosophy more than the other way around”5, as Jean-Paul Delahaye wrotein the early 2000s [DEL 00, p. 95]. In addition, we do not seek to lay out a pointless

1 J.-T. Desanti, preface to L. Brunschvicg [BRU 81, p. VI].2 See the title of the book by Kline [KLI 93].3 Toward the end of Chapter 7, we will be able to return to this and comment on the resultsobtained notably from the work of Gregory Chaitin.4 See, for example, Rashed [RAS 91]. G.-G. Granger has sometimes highlighted the reverse,as is the case with Leibniz, where the philosophical principle of continuity determines differentaspects of his mathematics. But this, in his own words, is an exceptional phenomenon [GRA 86].5 Further on in this book (pp. 95–104), the author lists different important philosophicalconsequences of the progress of Kolmogorov’s theory of complexity and, notably, the definitionof the randomness of a string as algorithmic “incompressibility”, which resulted in: (1) anew understanding of Gödel’s theorems of incompleteness; (2) an objective conception ofphysical entropy (Zurek); (3) Chaitin’s Omega number and the assurance of coherence inthe theory of measurement; (4) a new understanding of scientific induction, of Bayes’ rule

xiv Mathematics and Philosophy

culture. We only aim to communicate the essential. That is, in the teacher’sexperience, what is most easily lost or forgotten. The majority of this book will thusredemonstrate that philosophical reason, while it has undoubtedly been subject tomultiple inflexions over its history, can only be constructed by looking at thecorresponding advances made in science, and especially the discipline that containsthe major victories of the sciences: mathematics. From the Pythagoreans to thepost-modern philosophers, nothing of any importance has ever been conceived ofwithout this near-constant reference.

Implementing philosophy today assumes an awareness of this creative trajectory.Once this is done, there are, of course, still some evident problems: if we believe inour schema, then should today’s philosophy follow the same inspiration as thephilosophies of earlier ages? Is it possible for today’s philosophy to escape the biasesthat burden ancient systematic thinkers without denying its own nature? Whatdefinitive form must philosophy take today? These are but a few of the manyquestions that surround this reflection, which is, in our view, constantly inspired bymathematics. History has shown us that the true philosophers have not always beenthose who stirred up radical ideas, political criticisms or those short-sighted moralistswho, today, many consider great philosophers. This is chiefly due to their lack ofknowledge of science as well as the echo-chamber created by the media around themost insignificant things, which pushes the media itself to discuss nothing but thisphenomenon. However, the existence of real facts and strong movements, generallyignored by the media buzz, leads us to think that things of true importance arehappening elsewhere. Philosophy, with all due respect to Voltaire, used to besomething quite different. And, for those who are serious, this remains anundertaking that goes well beyond what we find today in journals and magazines.

A note on the notations used here: When we speak of the mathematics of antiquity,the Middle Ages or the Classical Age – in brief, the mathematics of the past! – we willuse present-day notations to ensure clarity. However, it must be understood that thesymbols that we will use to designate the usual arithmetic operations have only existedin their current usage for about three centuries [BRU 00, p. 57]. It was at the beginningof the 17th Century, for example, that the “plus” sign, (+) (a deformation of the “and”sign (&)) and the “minus” sign (–) began to be widely used. These symbols are likelyto have appeared in Italy in 1480; however, at that time it was more common to write“piu” and “meno”, with “piu” often being shortened to “pp”. In the 16th Century(1545, to be exact), a certain Michael Stiffel (1487–1567) denoted multiplication by

and Occam’s razor; (5) the distinction between random complexity and organized complexity(Bennett). A final epistemologically non-negligible consequence is the famous law propoundedby Kreinovich and Longpré, according to which if a mathematical result is potentially useful,then it is not possible for it to have a complex proof. It would seem to result from this that whichis complex is potentially useless, a result which many long-winded philosophers would do wellto contemplate (see [KRE 00] and [LI 97]).

Introduction xv

a capital M. Then, in 1591, the algebraist François Viète (1540–1603), a specialistin codes who used to transcribe Henri IV’s secret messages, replaced this sign by“in”. The present-day use of the cross (!) was only introduced in 1632, by WilliamOughtred (1574–1660), a clergyman with a passion for mathematics. The notation forthe period (.) owes itself to Leibniz (1646–1716), who used it for the first time in1698. He also generalized the use of the “equal to” sign (=). This was used by RobertRecorde (1510–1558) in 1557 but was later often written as the Latin word (æqualitur)or, as used by Descartes and many of his contemporaries, was abridged to a backward“alpha”. While the notation for the square root appeared on Babylonian tablets datingback to 1800 or 1600 B.C. (see Figure I.1), its representation in the form we knowtoday, (") dates back no earlier than the 17th Century. Its use in earlier mathematicsis, thus, only a simplification and has no historical value.

Finally, this book is not a history of mathematics, but rather a study of the impactof mathematical ideas on representations in Western Philosophy over time, with theaim of highlighting teachings that we can use today.

Figure I.1. The YBC 7289 tablet (source: Yale Babylonian Collection)

PART 1

The Contribution ofMathematician–Philosophers

Mathematics and Philosophy, First Edition. Daniel Parrochia.© ISTE Ltd 2018. Published by ISTE Ltd and John Wiley & Sons, Inc.

Introduction to Part 1

In antiquity, a period when science was both knowledge and wisdom, there wasno real distinction between a philosopher and a seeker of learning, that is, a personwho loved knowledge or loved wisdom. Thus, people studied and manipulated bothconcepts and quantities, which could be discrete (and, therefore, could be expressed inwhole numbers) or continuous (segments, surfaces, etc.). In Greece, as in virtually anysociety, the only numbers known from the beginning were whole numbers. However,the existence of division imposed the use of other numbers (fractions or fractionalnumbers) both to translate the form as well as the results of this operation. Initially,therefore, fractions were only ratios between whole numbers1.

It was the Pythagoreans who first created the theory of whole numbers and therelations between whole numbers, where they would sometimes find equalities (calledproportions or medieties). But, as they would very soon discover, other quantities existthat cannot be expressed using these numbers. For example, the Pythagoreans wouldexplore a spectacular and intriguing geometric quantity: the diagonal of a square.

Everyone knows what a square with a given side a is. The area of the square, S, isobtained by taking the product of one side by another. In this case, S = a ! a = a2.The Pythagoreans were interested in the diagonal of the square as they were trying tosolve a particular problem, that of doubling a square. In other words: how to constructa square whose area is double that of a square of a given side (a problem evoked inPlato’s Meno). The response, as it is well known, is that we construct the square thatis double the original square with diagonal d. But the question is: how is the length ofthis diagonal expressed?

1 Of course, as soon as there were real numbers, it was possible to think of relationsbetween them. At that moment, then, there would also be relations between irrational or eventranscendental numbers.

4 Mathematics and Philosophy

The Pythagoreans knew of a theorem, which we usually attribute to their leader,Pythagoras, but which is undoubtedly much older. The theorem states that, in anorthogonal triangle (that is, a right triangle), the square of the hypotenuse (thediagonal) is equal to the sum of the squares of the two sides of the right angle. If weapply this theorem to the square we considered above, we obtain:

d2 = a2 + a2 = 2a2

From this, it is easy to observe that d cannot be a whole number.

If we take a = 1, then d2 = 2. Thus, the number d is necessarily larger than 1,because if d was equal to 1, d2 would also be equal to 1. However, d must also besmaller than 2, because if d was equal to 2, then d2 would be equal to 4. This number,d, therefore, lies strictly between 1 and 2. However, there is no whole number between1 and 2. Thus, d is not a whole number.

In addition, as we will see further (see Chapter 1), we also prove that d cannot bea fraction or, as we say today, a “rational” number.

Here, we highlight the quantities that the Pythagoreans would, for lack of a betteralternative, define negatively. They called these quantities irrational (aloga, in Greek),that is, “without ratio”. The discovery of these incommensurable quantities or numberswould have large philosophical consequences and would require Plato, in particular,to completely rethink his philosophy.

Finally, as mathematics progressed, it was seen that certain numbers are thesolutions to algebraic equations but others could never be the solutions to equationsof this kind. These numbers, which are not algebraic (such as " or e, for example)would be called “transcendental”. They also brought specific problems with variousphilosophical consequences.

Greek geometry asked other crucial questions, such as those concerning thedoubling of a cube, the trisection of an angle (Chapter 2), or again the squaring of acircle. However, it found itself limited when it came to those constructions that couldnot be carried out using a scale and compass and which would not be truly resolveduntil the invention of analytical methods.

The squaring of a circle especially (Chapter 3) (i.e. how to relate the area of acircle and that of a square) would bring with it reflections on the infinite, thedifferences between a line segment and a portion of a curve, the contradictions linkedto the finite and the possibility of overcoming these contradictions in the infinite. Allthese speculations, as we will see, sparked off the reflection of Nicholas of Cusa.

The rational approximations of " – notably those given by Archimedes – wouldmobilize trigonometric functions, which were also used in astronomy to calculate

Introduction to Part 1 5

certain unknown distances using known distances. And the birth of financialmathematics, linked to the growth of capital, would play an important role in thediscovery of the logarithmic function and its inverse, the exponential function.

The progressive extension of calculations would then lead mathematicians tocreate new numbers. For example, from the Middle Ages onwards we have seen thata second-degree equation of the type ax2 + bx + c = 0 only admits real numbers asthe solutions if the quantity b2 # 4ac (the discriminant) is positive or nul. But whathappens when b2 # 4ac is negative? For a long time, it was stated that the equationwould have no solution. But then a subterfuge was invented that would make itpossible to find non-real solutions to this equation. A new set of numbers was createdfor this purpose – they were first called “imaginary” numbers and later “complex”numbers. These numbers are solutions to second-degree equations with a negativediscriminant.

Euler formulated an early law for the unification of mathematics by positing anequation that related the three fundamental mathematical constants: ", e and i (thislast being the fundamental symbol of the imaginary numbers). These numbers, whichwould later find application in the representation of periodic functions associated withphysical flux, would be the origin of a new representation of the world, where energyseemed to be able to replace matter. Bergsonian philosophy, as we will see, resultedfrom such an error.

Mathematicians have, over time, also invented many other types of numbers: forexample, ideal numbers (Kummer) or again the p-adic numbers (Hensel). We will notdiscuss them here as they are not very well known to non-mathematicians and thus tothe best of our knowledge have not yet inspired any philosophy.

1

Irrational Quantities

In traditional philosophy, what is the fundamental philosophical operation? It isthat which consists of constructing a miniature image of the world, as complete aspossible, and whose aim is to capture the essential of the real world. The benefit of thisapproach is patently obvious: simplify to understand better. That is, taking everythingtogether, we reduce so that we can retain more. We are, moreover, pushed to carry outsuch a task, which proves itself to be highly useful, for obvious reasons:

– there is, among other things, a vital necessity to know the relative importance ofeach thing, our own situation in the world as well as the place that is ours;

– let us also note that such a project is democratic;

– finally, everyone has the right to know who they are and where they are: the veryprocedure that allows this, in accordance with its objective, could only have emergedin a context that was conducive to its appearance (Ancient Greece).

In any event, this consists of factorizing all that is perceived into a certain numberof classes and then, for each group being considered, to choose one or more distinctrepresentatives1. The dimensional reduction, if it is correct, then becomes heuristicand leads to an undeformed model of the real world. But there are many ways ofoperating this concentration and, contrary to the old adage, in philosophy, alas, less,is not always more.

What then must be prioritized and preserved from the range of phenomena? Fromthis colorful variety, these often garish shades of existence that were made up, in thetime of the Ancient Greeks, by people, places and even the monuments around them –from all this, the earliest philosophers, the disciples of Pythagoras, only wished toretain the number.

1 Such a model has been described in detail in a publication [PAR 93a].



For them, the number was an able replacement for any thing as, with things andnumbers in the same stratification, it was easy to substitute one for the other.Numbers themselves could be contained within the first four (the Mystic Tetrad) and,consequently, the entire universe can thus be contained within the beginning of thenumerical series (1 + 2 + 3 + 4 = 10)2. How economical! Not only is the worldcondensed into symbols, but all we need is 4 of these to cover the full spectrum. Tothese thinkers, who were still naive, the alpha and omega in the real world werenothing but numbers and relations (logoi) between numbers. Reason itself, which isnothing but science being exercised, identified with these relations. It also bears thesame name (logos). Reason, therefore, is essentially proportion. At the time, reasonfell within the limits of the Pythagorean theory of medieties3. And, perhaps the thingthat is most difficult for us 21st Century people to understand, it is nothing else.

1.1. The appearance of irrationals or the end of the Pythagorean dream

Naturally, this kind of a perspective, rather too radical, would not be tenable inthe long run. A simple mathematical problem, the doubling of a square4, brought upthe first irrational, which we have named “square root of two” (

"2) and which,

geometrically, can be identified with the diagonal of the square on which isconstructed the square that is the double of a square with side one. As we have seen,the Pythagoreans named such quantities a-loga, that is, strictly speaking, “withoutratio” or “incommensurables”. It is easy to demonstrate why

"2 – or, as the Greeks

really called it “the number whose product with itself gives 2” (as they did not knowthe expression (

"a)) – is not rational. Any rational must be of the form p/q, an

irreducible fraction. But the property of an irreducible fraction is that its numeratorand denominator cannot both be even (if they were, we could of course furtherreduce this by dividing it by 2). Let us thus posit

"2 = p/q is irreducible. We then

have p2 = 2q2, which signifies that p2 is even, and therefore, p is even. Let us thenposit that p = 2n and substitute this value in the equation. We obtain 4n2 = 2q2, thatis q2 = 2n2, thus q2 is even, which signifies that q is even. We thus have acontradiction and

"2 is not rational.

2 This, according to M. Ghyka [GHY 31], is what the Pythagoreans called “Tetractys”.3 This theory is best known because of the texts of Archytas, Nicomachus of Gerasa and Theonof Smyrna (see [MIC 50]).4 The question of the appearance of incommensurables would provoke polemical debatesamong science historians. We have no precise trace for the discovery of irrationals in AncientGreek – only the accounts of commentators (Pappus, Proclus, Iamblichus, etc.), who wrotetheir accounts close to 700 years after the facts they were reporting. Pappus certainly tracesthis discovery to the Pythagorean sect, relating it to the question of the diagonal of the square,and attributes it to Hyppasius. Proclus, however, attributes it to Pythagoras himself. As forIamblichus, he considers that rather than the doubling of a square, this discovery of irrationalsarose from the problem of dividing a segment into extreme and mean ratios, that is the goldennumber.

Irrational Quantities 9

1.2. The first philosophical impact

We find many echoes of this discovery in Greek philosophy, especially in Platonicthought. In his weighty tome on Mathematical Philosophy [BRU 93], LéonBrunschvicg included the following observation:

“In Plato’s Dialogues, there is more than one hint that the discovery of irrationalsis not alien to the Platonic doctrine of science. In the introduction to Thaetetus, thedialogue that would mark the first degrees of analysis that went from perceptibleappearance to truth, Plato recalled the writings of his tutor, Theodore5, whoestablished the irrationality of

!5,

!7, etc. and pursued the search for irrational

square roots up to!17 6. In book VII of The Laws, he deplores, as a crime against

the nation, that young Greeks were left ignorant (as he was left ignorant) of thedistinction between commensurable quantities themselves and incommensurablequantities7, a distinction that he used as the basis for the ‘humanities’. Above all, theexample of Meno must be highlighted: the problem, one of the simplest of those thatcould arise after the discovery of incommensurability, consists of determining thelength of the side of a square that would be double that of another square with asurface of four feet. What is significant is the objective of this example: it was toprove the Reminiscence Theory of Knowledge. The Platonic Socrates introduces aslave who, it was claimed, without any direct learning, and using solely the effect ofnatural light which revealed itself, could find the veritable solution to the problem8.The first responses of the slave were borrowed from the framework of purearithmetic: the square with double the area seems to have a side with double thelength. But, double the length would be 4 and thus the doubled area would be 16. Theside of the square would, thus, be greater than 2 and smaller than 4, that is, 3. But thisresponse, which exhausts the truly numerical imagination, is still inexact: the squarewith a side of three feet would have an area of 9 feet. Socrates, thus, proposes anexclusively geometric reflection.

“Let the square be ABCD (Figure 1.1)9; we can juxtapose this with three equalsquares so as to obtain the quadruple area AEGF. Taking the diagonals BC, CI, IHand HB, we divide into two each of these four areas, equal to a primitive square. Thesquare BCIH is, therefore, double the primitive square; the side whose length would

5 Theodore of Cyrene, a mathematician who, according to Diogenes Laërtius (III, 6), taughtPlato mathematics [note by D. Parrochia].6 Thaeteus, 147d. See the study by [ZEU 10, p. 395 onwards].7 Plato, The Laws, 820c.8 Plato, Meno, 82b.9 Here, L. Brunschvicg goes back to [CAN 07, p. 217] [note by D. Parrochia].


be equal to"8 is the line that the Sophists call the diameter: it is from the diameter,

thus, that the doubled area is formed”10.

Figure 1.1. Meno’s square

Plato’s theory of science and, in particular, the idea of reminiscence are thus notanchored in mythology, as we believe only too often (appealing to the Myths is, asalways, simply a pedagogical or psychagogic tactic Plato uses to express himself) butinstead it is anchored in a rationality that is fundamental to the human mind; in thecognitive abilities of the mind which are expressed here, precisely, in the fundamentalmovement which, at the time of this “crisis” or “near-crisis” of the irrationals11, quitesuddenly saw the growth of an extension to the concept of the number.

10 Meno, 85b.11 The notion of “the irrational crisis” is contested today. Historians of mathematics tend tothink that the authors of the 19th Century, who described this period in history, overstatedthe importance of the “trauma”, being influenced themselves by the “set theory crisis” thatthey were living through. However, we can observe that the discovery of irrationals threw theGreek world into disarray, not so much within mathematics as in the external world, where the


1.3. Consequences of the discovery of irrationals

The Pythagorean discovery led to several important philosophical consequences.

1.3.1. The end of the eternal return

The end of “everything is numbers” was not simply the rejection of thePythagorean hypothesis according to which only whole and rational numbers existed.This idea had quite concrete consequences. It brought about a general modification ofthe cosmological representation of the world, especially the representation of time.Indeed, as Charles Mugler once wrote, it brought about a veritable “cosmic drama”.It marked the collapse of the Pythagorean concept of circular time where therevolutions of different heavenly bodies, assumed to be expressed only in wholenumbers, would give rise to the calculation of a lowest common multiple (LCM).They thus led to the Pythagorean concept of the Grand Year, at the end of whichperiod it was assumed that the heavenly bodies had returned to their initial positionand that life on earth, which depended on them, would recommence completely.12

However, the end of this periodic cosmology and the presence of possible disorder inthe celestial world were not the only consequences of the appearance of irrationals.13

1.3.2. Abandoning the golden ratio

Plato had already noted in classical Greek architecture that there were somedistortions between the apparent and real proportions in certain monuments, evenreproaching architects for having used falsehoods to get people the truth. Theproblem only worsened, as the Greek aesthetic shifted over time not only to anexcess of refinement and mannerism but toward a renouncement of reserve, sobrietyand equilibrium in favor of expressing a certain dramatic tension, a certain pathos orhubris of despair as in the famous group of Laocoon. This is a sculpture that presentspeople in a state of agony, muscles taut and bulging eyes with despair in their eyes, asthey are defeated by serpents. With this sculpture, which dates from after the 2ndCentury B.C., the Apollonian order was overthrown by much more troubling andtormented representations, which would, one day in the future, attract the German

consequences of this discovery would force aesthetics, morality and medicine to change theirview of the world, which was until then founded on the theory of proportions, that is, rationalnumbers alone.12 For more on this, see the preface by C. Mugler [MUG 69].13 Today, after the work of Kolmogorov (1954), we know, on the contrary, that the irrationalityof the ratio ! = T2/T1 of two periods T1 and T2 of two different celestial bodies of the solarsystem increases its stability. If ! is diophantian, the stability is still much better. This soundsthe end of the Pythagorean harmony.


Romantics14. This is nothing but another consequence of the collapse of the existingorder brought about by the undeniable existence of the irrational in mathematics.

1.3.3. The problem of disorder in medicine, morals and politics

As we know, Greek wisdom also recommended “nothing in excess”. Stobaeus,before Plato, already judged that there must be proportion in the soul and in the city:“Once the rational count is found, revolts will die down and amity increase” (Flor.IV, I, 139). Alcmaeon, although he certainly lacked the means of experimentallytaking measurements, tried to prove the equality of forces in the body [VOI 06,pp. 11–78]. According to a fragment gathered by Diels-Kranz, “Alcmaeon said thatwhat maintains health is the balancing of forces (tèn isônomian tôn dunaméôn),humid, dry, cold, heat, bitterness, sweet and others, the domination of any one ofthem (en autois monarchian) causing disease; this was because the domination of asingle is corrupting... health is the combination of qualities in the correct proportion(tèn summetron tôn oiôn krasin)”15.

With the appearance of the irrationals, the end of Greek life was, in the long run,programmed. Nothing would be in proportion anymore: not in the cosmos, nor in thehuman soul, nor in the city. The potential presence of a destabilizing element,introducing the incommensurable (irrational movement, unrestrained passion,tyranny, etc.) would, every time, threaten to bring about a rapid downfall.

It was, therefore, useful to put up the defenses and fight back – starting with thefield of mathematics itself.

1.4. Possible solutions

How does one rid oneself of these irrationals? Squaring them, that is, turning alogainto dunamei monon rêta, is an easy solution. However, this will, obviously, changethe value of the numbers. To remain faithful to the data given in the problem, we will

14 The Laocoon group strongly inspired a movement in German art and authors as diverseas Winckelmann, Lessing, Herder, Goethe, Novalis and Schopenhauer covered this in theircommentaries. In Winckelmann’s classic expression, which again inspired Lessing, Herder andGoethe, the Laocoon group illustrated the concept of the strength of the soul and the aestheticrule of tempered expressions. Contrary to this, however, we have Novalis. In line with Heinseand Moritz who saw this instead as “the most violent horror and the strongest emotion”, Novalissaw here the very sign of an aesthetics of excess (see [OST 03]). Much later, Spengler wouldsee in this sculpture the decline of Greek society and the end of the art and values of classicalantiquity.15 Fragment DK 24 B 4.


try to find approximation formulas that would make it possible to turn the irrationalsinto rationals.

One of the procedures, while it may not have been entirely known to thePythagoreans, was, nonetheless, anticipated by them. This was the procedure ofcontinuous fractions, which seems to have been explicitly introduced by the Hindumathematician Aryabhata (550–476 AD). This is written (using the modern forms)as:

"2 = 1 +

1

2 + 12+ 1

2+...

Today, we can easily obtain this formula in the following manner. Because 1 <"2 < 2, we first posit, as the first approximation:

"2 = 1 +

1

a, with a $= 0 [1.1]

From this, we then find the value for a. Hence:

a =1"2# 1

="2 + 1 [1.2]

However, since:

"2 = 1 +

1

a

upon replacing"2 in expression [1.2] by its value, we also immediately have:

a = 2 +1

a[1.3]

By then substituting this value for a in expression [1.1] and then in the successiveexpressions, wherever a appears, we obtain the desired formula.

It appears that we cannot find an explicit trace for continuous fractions in Greekmathematics earlier than the work of Aristarchus (3rd Century B.C.) and Heron (1stCentury A.D.). However, Paul-Henri Michel was able to suggest that the procedures todimidiate unity and other approximations that Thomas L. Heath was able to report16

contributed to anticipating them.

16 “Not only did the Pythagoreans discover the irrationality of!2; they demonstrated, as we

have seen, how to approach, as closely as we wish, their numerical value” (see [HEA 21,p. 167]).


Jules Vuillemin, following in their path, noted more recently that “thePythagoreans made use of infinite sets in their polygonal number tables and in thedefinitions of progressions” [VUI 01, p. 11]. Reflecting on the use of theseprocedures to demonstrate the irrationality of

"2, especially in the algorithms called

“Theon’s algorithms” and “alternate division”, he observed that we could thus easilydeduce the laws of continuous fractions from these [VUI 01, p. 71]. The Pythagoreanuse of triangular number tables also meant that it was not necessary to explicitlyknow these algorithms, but to know them, “only through certain properties of theirapproximation” [VUI 01, p. 71].

Despite their lack of resolution, according to Vuillemin himself, these proceduresseem to have served as models for the Platonic method of division. In his own words,again, “while logical rigor is lacking in this initial recourse to finite sets, and whilethese difficulties inherent to continuous fractions also affect their rudiments, let usremember that the chief obstacle Greek mathematics came up against is the idea ofthe real number and we will see Theodore conceiving of roots of natural, non-squaredwhole numbers as the limits of the infinite series of rational approximationx [VUI 01,p. 106].

1.5. A famous example: the golden number

Among the ratios that the Pythagoreans loved, one could pass for a clevercompromise: this was a ratio and, at the same time, corresponded to an irrationalquantity. This ratio is defined in the following manner. We posit:

a+ b

a=

a

b

But this is equal to:

1 +b

a=

a

b%& a

b+ 1 = (

a

b)2 %& (

a

b)2 # a

b# 1 = 0 [1.4]

We then posit:

k =a

b

And equation [1.4] becomes:

k2 # k # 1 = 0

One of these two solutions (the positive solution) to this new equation (whichwould, moreover, find many applications in the field of aesthetics, notably


architecture) has been the subject of reams of writing over history17. The solutionsare classically obtained as follows. The discriminant of the equation is:

! = b2 # 4ac = 5

which gives, as the roots,

k! =1 +

"5

2, k!! =

1#"5

2k! is conventionally called the “golden number” or the “golden section”. We ordinarilydesignate this by the letter !.

We then observe that k!.k!! = #1, and that k! + k!! = 1. From this, we then canwrite:

#k!! =1

k!=

1

1# k!!=

1

1 + 1k!

Hence, the rational approximation of the “golden number”:

! =1 +

"5

2= 1 +

1

1 + 11+ 1

1+...

By developing the successive approximations of !, we then have the set of ratiosformed by the numbers belonging to a famous mathematical series18:

1

1,2

1,3

2,5

3,8

5,13

8,21

13,34

21,55

34,89

55, etc.

17 It must be noted that since the work of A. Zeising [ZEI 54], in the mid-1800s, and ofM. Ghyka [GHY 31], in the first half of the 20th Century, the golden number has met withmuch success. While this study was able to stimulate some research (see [CLE 09, pp. 121–123]), it must be admitted that a certain number of errors were also propagated through itand that the omnipresence attributed to it in nature or art was often mythical (see Neveux andHuntley’s critical study [NEV 95]). Generally speaking, the work on this subject – which mostoften repeats itself – flourishes. We can cite, among many others, Herz-Fischler [HER 98].The mystical aspect of the number seems to have been emerged in the 19th Century with thetranslation of Pacioli [PAC 80]. The expression “golden section” does not seem to date backfurther than the 19th Century and Martin Ohm’s work.18 The series of numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc., discovered by the Italianmathematician Leonardo Fibonacci in the 13th Century, is characterized by the fact that eachnumber, starting from two, is the sum of the two previous numbers in the series. This serieshas its origins in a question that Fibonacci asked in his Liber Abaci in 1202: “How many pairsof rabbits can be produced in a year from a single pair if each pair produces a new pair everymonth, each new pair reproducing from the second month?” The first numbers in the seriesrespond to this question. The formal definition of the Fibonacci series, given by Kepler in the15th Century, is:

F (1) = F (2) = 1 F (n) = F (n" 1) + F (n" 2)


Knowing that the above ratios may be denoted in a generic manner by F (n+1)F (n) , we

can demonstrate that:

limn"#

F (n+ 1)

F (n)= !19

1.6. Plato and the dichotomic processes

The processed called the “dimidiation” of unity, a characteristic of the goldennumber, finds an obvious parallel in language with the dichotomic processes, whichPlato would use increasingly often in his dialogues. In fact, Plato compared ideas tonumbers and a wrong calculation to an identification error (Thaetetus, 199c). Andlike the Pythagoreans, who saw a simple correspondence between numbers andthings, Plato believed for a long time that it was possible to establish a simplecorrespondence between the intelligible and the tangible, ideas and their referents inthe world. This belief, however, threw up many difficulties, as can be seen in theconversation between the young Socrates and Parmenides in the eponymousdialogue.

The Parmenides, and then the Sophist, acknowledged this failure and the laterdialogues raised other objections. Thus, the Philebus showed that certain ideas, such asthose of pleasure, which had multiple variables and that could only be attained throughexcess or failure, could be difficult to identify with intangible, full or whole realities.Irrational numbers are like this: they cannot be used in clearly defined relations andare not commensurable.

To approach them, we must have a process analogous to the dimidiation of unity.The processes that play this kind of a role are dichotomic processes (or proceduresfor binary divisions in language). Thanks to such “algorithms”, the definitions ofconcepts, which are no longer considered in a granular manner like whole numbers,but rather as a mixture of the limited and unlimited that is, like irrationals, may beapproached through successive divisions that converge on a final, stable element.These procedures, which originated in the Gorgias (450a), and had become widelyused in dialogues that followed the Parmenides, are the exact transposition ofcontinuous fractions or, as we have said, algorithms that anticipated these20.

19 Botanists consider that the ratios given express the most common phyllotaxis. Furthermore,a mathematical study has also made it possible to suggest that there may be some relationbetween the Fibonacci numbers and the logarithmic spiral – a curve that is found in nature[JEA 78, p. 75; ARC 94].20 The posterity of continuous fractions is remarkable. They were generalized and applied tothe approximate calculation of " and e and were recently used in the theory of dynamic systems(see [YOC 06, pp. 403–437]). A continuous fraction’s formula, discovered by Ramanujan andthat he proposed to Hardy in 1913, even introduces a relation between e," and #.


1.7. The Platonic generalization of ancient Pythagoreanism

1.7.1. The Divided Line analogy

Let us now consider Plato’s extremely famous text from the Republic (VI, 509e–511e), called “The Divided Line analogy”, in which Plato describes the world bycomparing it to a line segment that he constructs as follows:

“Now take a line which has been cut into two unequal parts, and divide each ofthem again in the same proportion, and suppose the two main divisions to answer, oneto the visible and the other to the intelligible, and then compare the subdivisions inrespect of their clearness and want of clearness, and you will find that the first sectionin the sphere of the visible consists of images. And by images I mean, in the firstplace, shadows, and in the second place, reflections in water and in solid, smooth andpolished bodies and the like: Do you understand? – Yes, I understand! – Imagine,now, the other section, of which this is only the resemblance, to include the animalswhich we see, and everything that grows or is made. – I can imagine it, he said. –Would you not admit that both the sections of this division have different degrees oftruth, and that the copy is to the original as the sphere of opinion is to the sphere ofknowledge? – Most undoubtedly! – Next proceed to consider the manner in whichthe sphere of the intellectual [noetic] is to be divided. – In what manner? – Thus:There are two subdivisions, in the lower of which the soul uses the figures given bythe former division as images; the enquiry can only be hypothetical, and instead ofgoing upwards to a principle descends to the other end; in the higher of the two, thesoul passes out of hypotheses, and goes up to a principle which is above hypotheses,making no use of images as in the former case, but proceeding only in and through theideas themselves”.

A line is, thus, divided into two unequal parts which are, in nature, visible andintelligible. Each section is then divided in the same ratio. This, which is visible,relative to a relation of clarity and obscurity, makes it possible to confront real bodies(animals, things) with their images (shadows and reflections of these bodies,sometimes called “simulacra”). With regard to the intelligible segment, this is alsodivided into two parts: on one side we have ideas (in the Platonic sense of the term:idea of beauty, idea of justice, etc.) and mathematical objects.

Figure 1.2. Plato’s line, from the Republic


Before any philosophical interpretation, we must understand what exactly Platohas done and why he divided this line in this manner. On the one hand, we have

AC

CB=

AD

DC=

CE

EBhence : DC =

AD.CB

AC[1.5]

On the other hand, we have

AC

AB=

AD

AC=

CE

CBhence : CE =

AD.CB

AC[1.6]

As we can see, from [1.5] and [1.6], two central segments in Plato’s lines areequal21. In other words:

DC = CE [1.7]

The most plausible interpretation of this situation is that if these two centralsegments represent objects from the tangible world and mathematical ideals, then theequality suggests that mathematics can be rigorously applied to the tangible world. Inthe tangible world, it is the world of living beings and real things. The problem thatthen arises is knowing whether or not images (or what Deleuze could call, using aninappropriate term, “simulacra”) can be compared to rationality. This was the grandproblem that Plato would work on till the end of his life. His final philosophy, thetheory of “ideas and numbers” (which we know through Aristotle) replied in theaffirmative: the world, from the purest ideas to the most mixed tangible complexities,is a hierarchy of mixed objects that inform tangible reality, even if less and lesseffectively. But nothing in the world is entirely lacking in reason.

1.7.2. The algebraic interpretation

Let us say that k is the ratio between the line and its subsegments.

It is then remarkable that the configuration of Plato’s division makes it possible tohave any kind of ratio (or almost) k.

Let us posit:

AC = a, AD = a1, DC = a2

CB = b, CE = b1, EB = b2

21 This result was known to authors of that period and, notably, to commentators on Plato’swork.


Plato’s hypotheses were:

a = a1 + a2b = b1 + b2

b = ka, b2 = kb1, a2 = ka1

We can then write:

b = ka & b1 + b2 = k(a1 + a2) & b1 + kb1 = a2 + ka2 & b1(k + 1) = a2(k + 1)

And for any k $= #1:

a2 = b1 or, as we saw earlier, DC = CE

1.7.2.1. Impossibilities

A case where k = #1, which assumes the existence of negative numbers andoriented segments, corresponds to a situation that Plato could not imagine but that canbe considered in algebra. This, however, leads to an impossible line. Similarly, a casewhere k = 0 also makes the line and its segments impossible. In any other situation,including when k = 1 (divided equally22), Plato’s hypotheses remain valid. This is,therefore, a very general situation.

1.7.2.2. The case where k = !

Let us now consider – one example among many others, but perfectly compatiblewith Plato’s hypotheses – a case where the ratio k represents the golden number itself.For this, it is enough to posit that

k = !

Among all possible ratios that define the division of a line AB into two segmentsAD and DB, there is, in fact, one where the ratio of the smaller segment to the largersegment is the same as that of the larger segment to the undivided line. This preciselydefines the ratio associated with the “golden number”. We have

! =AD

DB=

DB

AB

22 Thus, the text being ambiguous over whether we should read “isos” or “anisos” (equal orunequal) for the division of the line does not influence the result of the operation in any way.


This is the characteristic ratio of the golden section and Plato’s line can be dividedin this ratio (among others) and this ratio accorded a significant importance by thePythagoreans, being the key aspect governing the construction of the famous pentacleof the Pythagorean society.

In Figure 1.3, we have the following ratios:

BE

DE=

DE

FE=

FE

JF=

JF

OF=

OF

ON

Figure 1.3. The pentacle of the Pythagoreans

By dividing his line as he did, Plato thus retained the general principle ofharmony, but extended this harmony to any kind of ratio. We can, thus, say that hequite considerably generalized Pythagorean philosophy23.

1.8. Epistemological consequences: the evolution of reason

Based on the facts we have seen, it may be said that the evolution of scienceperiodically requires revising reason, which is not ingraved in stone, independent of

23 Here, we diverge from J. Vuillemin’s interpretation [VUI 01, p. 89], which assumes that theratio in which the line is divided in the Republic is the golden number. We cannot see anythingin this study that would validate such an interpretation. On the contrary, admitting that this ratiois one of the possible ratios is absolutely exact, and admitting that the line may also be dividedin this ratio poses no problem.


the activities through which it works and which oblige it to be transformed. GastonBachelard noted this in a study that is a perfect summary of our opinions:

“In sum, science instructs reason. Reason must obey science, the most evolvedscience, the evolving science. Reason has no right to overvalue an immediateexperience; on the contrary it must be balanced with the most richly structuredexperience. In all circumstances, the immediate must give way to the constructed.Destouches often repeats: if arithmetic, in distant developments, is revealed tocontradict itself, we shall reform reason to wipe out the contradiction and preservearithmetic and keep it intact. Arithmetic has demonstrated so many instances ofefficiency, exactitude [and] coherence that we cannot think of abandoning itsorganization. Faced with a sudden contradiction, or more exactly, faced with asudden necessity for a contradictory use of arithmetic, there would arise the problemof a non-arithmetic, a panarithmetic, that is a dialectical extension of intuitionsregarding numbers that can make it possible to span both the classic doctrine as wellas the new doctrine” [BAC 40, p. 144].

We see here that the Pythagorean reason, which to begin with was essentiallyproportion, ratio, coincides, at this period, with its use within fundamentalmathematics. In order to evolve, it was necessary that mathematics also evolve. Withthe aloga, the Greeks discovered that there are numbers and reason beyond what theyhad held, so far, as being reason (logos as proportion). They thus needed to evolvetheir conception of reason and admit the incommensurable. Mathematically, thissituation would lead directly to analysis: it not only led the way out of Greekgeometry but required the discovery of another type of rationality than the rationalityof proportion, thus opening the door to something that challenged reason withinitself, namely, not only folly but anything that would, in general, disturb humanreason: the infinite, the sublime, asymmetry, etc.

What was also revealed by the “crisis” or “pseudo-crisis” of irrationals is thatthere is, therefore, no single reason from the start to the finish of history. Here again,Bachelard, has clearly appraised the phenomenon. Thus, in Le nouvel espritscientifique (The New Scientific Spirit), he observes that the revolutionary growth ofscience “must have a profound reaction on the nature of the mind” and that “the mindhas a variable structure from the time that knowledge has had a history”. There is,therefore, irreversible progress in the development of scientific knowledge, which isenough to distinguish science from the other modes of understanding the real world:

“Human history may, in its passions and prejudices, in everything that is born ofimmediate impulses, be an eternal beginning; however, there are thoughts that do notstart over; these are thoughts that have been rectified, broadened, completed. They donot go back to their restricted or faltering beginnings. Moreover, the scientific spirit isessentially a rectification of knowledge, a broadening of the framework of knowledge.It judges its past and condemns it. Its structure is the awareness of its historical faults.


Scientifically, we think of the truth as the historical rectification of one long error, wethink of experience as the rectification of the common and primary illusion. All theintellectual life of science plays, dialectically, on this differential in knowledge, on thefrontiers of the unknown. The very essence of reflection is to understand that we havenot understood” [BAC 73, pp. 173–174].

Reason, therefore, evolves based on science. We do indeed say “based onscience” and not on god knows what other discourse or activity (theological,philosophical, literary, poetic, etc.) whose transformational virtue is not proven. Onthe other hand, science and technology, which confront true realities, and especiallymathematics, which encounters the most formal aspect of reality, have unequalledpower to transform. With no offence to certain sociologists (who wish to considerscientific discourse as nothing more or less than any other social product, such as artor religion), it is science that forces the most radical transformations and that setsthem into motion itself24. With science, borders that had seemed insurmountablemelt away, like the fragile constructions of charlatans of the mind. After amathematical revolution, we can no longer think like we did before.

24 It is possible that art (when not retrograde, “pompier” (The French term for Academic art), orpurely negative, as it is often is today) can anticipate these transformations; religion, however,trails far behind and tries, at best, to “limit damages”.

2

All About the Doubling of the Cube

After the problem of doubling a square, the second problem confronting Greekmathematicians was the doubling a cube.

This problem consisted of asking how to construct a cube whose volume wasdouble that of a cube with a given side.

To all appearances, this is a simple transposition of the “planar” square probleminto three dimensions. In reality, the problem of doubling a cube (with its problemof trisecting an angle) with that of squaring a circle is one of the three mainproblems posed by the Greek geometricians, marking the limits of their knowledgeand consequently bringing them, so to speak, closer together with our knowledge. Ineffect:

– finding double the volume of a cube with a given side is the same as determiningthe cube root of two;

– this operation is impossible to carry out using a scale and compass;

– by seeking (and finding) solutions using intersection of cones, cylinders and tori,the Greeks would discover conic sections (parabolas and hyperbolas, notably) as wellas non-trivial curves such as conchoids and cissoids. They thus reached the frontiersof their geometry, beyond which all turned to analysis.

Moreover, as we will see, this research also left its traces in philosophy andaccording to Plato, the harmonization of the universe and society required theresolution of “solid” problems. Here again, philosophy found itself at the center of aseries of heated debates.



2.1. History of the question of doubling a cube

The origin of this problem is as follows: the legend begins in Delos, an island inthe Cyclades archipelago and the birthplace of Apollo. According to Eratosthenes, theking, Minos, wanted to build a monument to his son, Glaucus. He found the tomb thatthe architect had built too small and asked him to double the size of the tomb. The poetbelieved, moreover, that this would simply consist of doubling each side – clearly anincorrect solution1. Another version of this tale is found in a tragic poem that evokesthis legend, where it is said, “Too small is the tomb you have marked out as the royalresting place. Let it be twice as large, with no error”.

According to von Wilamowitz, the famous philologist, this poet was neitherAeschylus nor Sophovolumecles nor Euripides, but instead it was some obscure poetwhose only claim to fame is this problem.

2.2. The non-rationality of the solution

Let x = 1 be the side of the cube. We must find x3 = 2.

If rational solutions exist, then they take the form: x = pq . By replacing x in the

above equation with its value, we have:

p3

q3= 2

Hence

p3 = 2q3

This equation is evidently problematic. It is clear, as we can quickly demonstrate,that there is no rational solution.

2.2.1. Demonstration

It is clear that a cube is a number multiplied by itself three times. These threefactors are the primary factors. Thus, the number of primary factors of a cube (let uscall this p3) is necessarily divisible by three. On the contrary, for 2q3, which is the cubemultiplied by 2, the number of primary factors is no longer necessarily divisible bythree. This means that the equality can no longer be satisfied. The number x, therefore,cannot be rational and it is not possible to double the cube by construction.

1 If we do double each side of the cube, as the poet suggests, we would have a cube whosevolume v, instead of being equal to a3, would be (2a)3 , that is, 8a3: thus, a cube that is eighttimes larger!

All About the Doubling of the Cube 25

2.2.2. The diagonal is not a solution

We can see, furthermore, that 3"2 is not constructible2 and that the solution of the

diagonal, which held well for the doubling of a square, cannot be applied here.

Contrary to the case of the square, the use of the diagonal does not yield thecorrect result. Let D be the largest diagonal of the cube; d = a

"2 is the diagonal of

the square that forms its base. In accordance with Pythagoras’ theorem, we have:D2 = d2 + a2 = 2a2 + a2 = 3a2. Hence, D = a

"3. If the volume of the initial cube

is a3, the volume of this new cube will then be equal to D3 = 3a3"3, that is, more

than thrice the initial volume. Using the diagonal is not, thus, a possible solution.

2.3. The theory proposed by Hippocrates of Chios

For a long time, geometricians were unable to make any headway with thisproblem until Hippocrates of Chios demonstrated that the problem could be reducedto another. This problem consisted of finding two proportional means, in acontinuous proportion, between two given lines.

Hippocrates used the following steps (see Figure 2.1).

Let a be the side of a cube, v1 be the volume of the initial cube and v2 the volumeof the doubled cube. We have:

v1 = a3 v2 = 2a3

Let us then consider a parallelepiped rectangle, formed of two initial cubes, and,consequently, whose total volume is 2a3. Let us replace this parallelepiped rectanglewith a parallelepiped with length a, breadth x and height y.

We obtain the equation:

axy = 2a3

Hence,

xy = 2a2

That is,

a

x=

y

2a

2 The non-constructibility of the square root of two was demonstrated only by Pierre-LaurentWantzel (1814–1848) in 1837.


a

a

a2a

x

x

x

x

y

Figure 2.1. The procedure used by Hippocrates of Chios

But in the case where the parallelepiped is equal to a cube of side x, we again have:

axy = x3

which means that:

x2 = ay ora

x=

x

y

hence, finally:

a

x=

x

y=

y

2a[2.1]


2.4. A philosophical application: platonic cosmology

This solution, which consists of introducing two proportional means between twoquantities, was used by Plato in Timaeus (31c–32c) to harmonize the form of theworld. Let us consider the following text:

“Now, that which is created is of necessity corporeal, and also visible andtangible. And nothing is visible where there is no fire, or tangible whichhas no solidity, and nothing is solid without earth. Wherefore also God inthe beginning of creation made the body of the universe to consist of fireand earth. But two things cannot be rightly put together without a third;there must be some bond of union between them. And the fairest bond isthat which makes the most complete fusion of itself and the things whichit combines; and proportion is best adapted to affect such a union. Forwhenever in any three numbers, whether cube or square, there is a mean,which is to the last term what the first term is to it – and again, when themean is to the first term as the last term is to the mean – then the meanbecoming first and last, and the first and last both becoming means, theywill all of them of necessity come to be the same, and having become thesame with one another will be all one. If the universal frame had beencreated a surface only and having no depth, a single mean would havesufficed to bind together itself and the other terms; but now, as the worldmust be solid, and solid bodies are always compacted not by one meanbut by two, God placed water and air in the mean between fire and earth,and made them to have the same proportion so far as was possible (asfire is to air so is air to water, and as air is to water so is water to earth);and thus He bound and put together a visible and tangible heaven. Andfor these reasons, and out of such elements which are in number four, thebody of the world was created, and it was harmonized by proportion, andtherefore has the spirit of friendship; and having been reconciled to itself,it was indissoluble by the hand of any other than the framer”.

What is Plato saying in this text?

1) When we have two separate elements, a third term is needed to relate them. Wethus have a geometric proportion of the type:

a

b=

b

c

that is to say, the relation between the first and second term is the same as that of thesecond to the third term. We can, moreover, express this proportion in different ways.For example, by permuting the extremes, that is, by writing:

c

b=

b

a


or again, by reversing the two ratios:

b

a=

c

b

This type of ratio corresponds to the ratios of constructible numbers, which giverise to planar constructions using a scale.

2) Plato then shows that the problem of harmonizing the universe does not fit intothis framework as it is a “solid” problem, a volumetric one, and is, therefore, morecomplex. He says that we must have two proportional means to harmonize solids andto define the general harmony to which the universe’s form must correspond. He hadalready postulated the existence of this phenomenon in Gorgias (508a):

“And wise men assure us, Callicles, that heaven and earth, gods and menare held together by communion and friendship, by orderliness,temperance, and justice; and that is the reason, my friend, why they callthe whole of this world by the name of order, not of disorder ordissoluteness. Now you, as it seems to me, do not give proper attentionto this, for all your cleverness, but have failed to observe the great powerof geometrical equality amongst both gods and men”.

This harmony, which will be established in the cosmos, thus establishes twoproportional means (air and water) between two extremes (fire and earth) such thatwe finally have:

FireAir

=Air

Water=

WaterEarth

How do we explain this proportion and its “solid” nature? One answer is to lookat things from an historical perspective.

It can be considered that this proportion was already outlined in Phaedon (111ab)where Plato shows that what ether is to air, air must be to water. We thus havesomething like:

EtherAir

=Air

Water

Banquet postulated, furthermore, that there existed intermediaries between Godand men, called daïmônes or demons. Plato uses the language of geometry or musicthroughout to suggest the function of the liaison that these demons carry out as he saysthat in the medium term (en mesô) they fill (sumpleroi) the God–man interval.


We must then find a place for these “demons”. If, as is plausible, we range themalongside the intermediary region of air and water, leaving ether for divinity, we mustthen find another place for humans. We, thus, think of the water–earth region.

Whatever it may be, the problem of this double harmonization obviously brings tomind the translation, of Hippocrates of Chios, of the problem of doubling a cube. Thisis especially true if we know that the geometric element related to the earth is indeedthe cube.

We will also see that for Plato, geometric proportion not only played a part inthe universe but also in the city. We can cite here the text in Laws (VI, 757b), whichcontrasts two types of equality, one founded on chance, and the other on geometry.As in III, 693d, Plato seeks for a middle term between two extreme political regimes,the rule of one (or a monarchy) and the rule of the people (or democracy). He thusdefends, for men, the idea of distributive justice, which consists of greater advantagesto those who are better and fewer advantages to those who have less merit. In otherwords, he defends – as he says himself – a justice that “attributes to all parties the sharethat is appropriate to them” (Laws, VI, 757c). However, as we can observe, in politicshe returns to a single proportional mean, which may seem a very weak solution. Thequestion that then arises is that of knowing what exactly philosophy knew about realsolutions to the problem of doubling a cube.

2.5. The problem and its solutions

To learn about these solutions, let us review the history of the problem and theprogressive quest for a solution.

Having already discussed the legend of Minos, let us now look at a second legendwhere this problem is evoked. According to another tradition, it is said that around430 B.C., the residents of Delos were struck down by disease and wished to put anend to the plague that was laying waste to their country. The Oracle told them thatthey must double the size of the altar dedicated to Apollo. After several attempts thatended in failure, the plague redoubled in intensity and the citizens of Delos went insearch of Plato to seek his advice.

2.5.1. The future of the problem

Plato, unfortunately, did not find the geometric solution to the problem (as there isno solution using a scale and compass). He claimed, however, that the Oracle had notmeant that the god wanted an altar that was twice the volume of his current altar;instead, by posing this problem he wished to shame the Greeks for neglectinggeometry. He is then likely to have sent the Delians onto Eudoxus and Helikon.


We know, however, that this problem was later studied, if not by Plato himself,then by one of his disciples. This is because the “mechanical” solution is – wrongly!– attributed to Plato, even though it was actually propounded by a young geometricianamong his followers (we will return to this solution later).

In any case, this problem of the doubling of a cube would be passionately takenup by Greek geometricians, who proposed several different solutions. We will look ata few of these. It must also be known that this question was later studied by Descartes(1637), and then by Gauss, before the French mathematician Wantzel (1814–1848)demonstrated, in 1837, that this problem, like that of the trisection of an angle, hadno solution that could be arrived at using a scale and compass. In the Ancient Greekperiod, the problem was studied by the greatest geometricians of the time: Archytas,Eudoxus, Menaechmus, Nicomedes and, later, Appolonius, the inventor of the conicsection.

2.5.2. Some solutions proposed by authors of the classical age

As solutions were not constructible using a scale and compass, authors necessarilyhad to either trace freehand analytical curves, or use mechanical means to constructthem. This is how two sets of solutions, analytical and mechanical, were born.

2.5.2.1. Mechanical solutions

Among the mechanical solutions, we will examine only the Platonic(pseudo-Platonic) solution here, as reported by Theon of Smyrna in his work titledAn Exposition of Useful Mathematical Knowledge for Reading Plato (publishedtoward 150 A.D.).

The problem of doubling the altar, such that the second altar would be similar inform to the first, leads to the doubling of the cube of an edge. As we saw earlier,Hippocrates of Chios found that if we insert two continuous proportional means, xand y, between the side of a cube, a, and the double of this side, 2a, then the firstmean, x, is the side of the doubled cube. We thus have by definition:

a

x=

x

y=

y2

a

a3

x3=

axy

2axy=

1

2hence: x3 = 2a3

One of the members of the academy is likely to have resolved the problem of twoproportional means using an instrument formed of two rules, KL and GH. One ofthese, being mobile and parallel to the other, which was fixed, would slide betweenthe grooves of two jambs, FG and MH, fixed perpendicular to this (see Figure 2.2).The details of the construction are not important. This kind of solution is mechanicalas it requires the use of an instrument other than a scale and compass. We learnt of this


through Eutocius of Ascalan, a geometrician of the 3rd Century A.D., in a commentaryon book II of the Treatise on the Sphere and Cylinder by Archimedes3.

Figure 2.2. The “mechanical” solution proposed by the academy

As we may well suspect, it is highly unlikely that this kind of a solution wasproposed by Plato himself, as the kind of construction used here was not thephilosopher’s preferred kind. All that we can say is that the existence of such asolution proves that the problem of doubling a cube was known to the academy andwas a topic of much debate within it.

2.5.2.2. Analytical solution2.5.2.2.1. Archytas’ solution

The oldest (dating back to the first half of the 4th Century B.C.) and most elegantsolution is, undoubtedly, that proposed by Archytas. This solution determined the

3 Let a and b be the two lines between which we wish to insert two proportional means. Wetrace two perpendicular lines AE and CD, on which we take, from their point of intersection,AB = a and BC = b. We then apply the instrument to the figure such that the edge of onescale passes through the point A and the edge of the other passes through the point C. We thenmore or less separate the mobile scale from the fixed scale and, at the same time, we can turnthe instrument through the plane of the figure, such that the edges of these two scales alwayspass through the points A and C. The extensions of the lines AB and BC pass through thevertices of the rectangle followed by the instrument at the same time. As the two triangles,ADE and CDE, are right triangles, the height of each of them is a proportional mean betweenthe segments of the hypotenuse and we have:

AB

CD=

BDBE

=BECD

Thus, BD and BE are proportional means between AB = a and BC = b.


point that made it possible to calculate the desired line as the intersection of threesurfaces, that of a cone, a cylinder and a torus.

Archytas began by reformulating the problem posed by Hippocrates of Chios; thatis, substituting the problem of doubling the cube with the problem of constructing twoproportional means x and y, between two quantities a and 2a, as in formula [2.1].From here onward, he introduces two novel concepts:

1) He generalizes the problem of doubling a cube, stricto sensu, to the moregeneral problem of finding two proportional means between two quantities, whateverthe ratio between them. In other words, we must find x and y such that:

a

x=

x

y=

y

b[2.2]

(in this case, b is no longer necessarily equal to 2a)

2) Archytas was aware of the three-dimensional nature of this problem (as itinvolved the doubling of a cube) and he went beyond the planar view and, from theoutset, worked in the geometry of space. His solution, passably complicated andwhich we will not present in detail here, consisted of bringing in the intersection ofthe three surfaces mentioned earlier (cone, cylinder and torus) in an extremelyremarkable construction (see Figure 2.3).

Figure 2.3. Archytas’ solution

Expressed in the language of analytical geometry, this gives us the followingtranscription. Let AC be the x axis. The y axis will be the line passing through A andperpendicular to the plane ABC. The z axis is the line passing through A andparallel to PM .


The point P is completely determined by the intersection of the following surfaces:

x2 + y2 + z2 =a2

b2x2 (cone) [2.3]

x2 + y2 = ax (cylinder) [2.4]

x2 + y2 + z2 = a!x2 + y2 (torus) [2.5]

hence AC = a,AB = b.

From [2.4], we can derive:

(x2 + y2)2 = a2x2 hence:(x2 + y2)2

b2=

a2x2

b2

and thus, by substituting this in [2.3], we obtain:

x2 + y2 + z2 =(x2 + y2)2

b2[2.6]

From [2.6], we can then derive:

!x2 + y2 + z2!

x2 + y2=

!x2 + y2

b[2.7]

and from [2.5], we deduce:

a!x2 + y2 + z2

=

!x2 + y2 + z2!

x2 + y2[2.8]

This finally gives us:

a!x2 + y2 + z2

=

!x2 + y2 + z2!

x2 + y2=

!x2 + y2

b[2.9]

or, by substituting the segments for their values

AC

AP=

AP

AM=

AM

AB


or, again

AC.AB = AM.AP

Thus, the cube with the side AM is to the cube with side AB as AC is to AB. Inparticular, when AC = 2AB, we have AM3 = 2AB3.

2.5.2.2.2. Archytas’ curve

We will then observe that [2.5] can be rewritten in the form:

(x2 + y2 + z2)2 = a2(x2 + y2) [2.10]

This equation [2.10], along with [2.4], forms a system that describes theintersection of a torus with a null hole and of a cylinder of revolution with axisperpendicular to the central circle of a torus and of the same radius as this circle. Thisintersection corresponds, in fact, to a particular curve, which is called “Archytas’curve”. This involves a 3D biquartic, which can be expressed in Cartesianparametrization as follows:

x = a cos2 t y = a cos t sin t z = ±a!(1# cos t) cos t # "

2' t ' "

2

Figure 2.4 presents this strange curve, which is called Archytas’ curve.

Figure 2.4. Archytas’ curve


Figure 2.5 shows the form of this curve and torus–cylinder intersection in space.

Figure 2.5. Archytas’ curve in 3D-space

2.5.2.2.3. Reflections on Archytas’ procedure

As we know, in this period (antiquity, 4th Century B.C.) where reason –constructed by Pythagorean mathematics – was based on the study of numbers andratios between numbers, establishing proportional ratios between the objects beingstudied was the only usable method. This was true both for investigations in nature aswell as moral reflection or political practice.

However, where Plato only saw relatively simple proportions established betweenthings, Archytas, in order to define the elements in his way, had to know aboutnon-trivial intersecting surfaces (cone, cylinder, torus) situated in 3D space.

The definition of invariants (here, proportional ratios between two distantelements that seem to have no relation, through the intermediary of two otherelements) was accompanied by an abundance of theories that led mathematicians tothe borders of Greek geometry. Beyond, they caught a glimpse of the marvels of anunknown universe where the stability of the simplest form was related to figures withextremely tortuous contours, which were themselves plunged into a space of higherdimension.

2.5.2.2.4. A possible political application

This kind of theorizing, which would only be reinforced over time, gave Archytasa place in a sequence of events that led directly to the most elegant accomplishments


of the 20th Century. It proved again that in order to harmonize bodies and souls, asPlato wished to do, to weave together different human temperaments or pacify entirecities, it was not enough to write down trivial medieties.

Establishing proportional means between two extremes requires moving intospaces of higher dimensions and, in these spaces, tracing trajectories of intersectionsthat are not at all evident. Politics, as an art of interlinking, often takes tortuousforms; as tortuous, no doubt, as the Archytas curve. Was Plato truly aware of this?We can allow ourselves this speculation.

However, it is possible, after all, to compare the Archytas curve to what thephilosopher said about the art of politics, which resembles the art of the weaver andconsists of blending temperaments as the weaver blends the warp and weft. Theincredibly beautiful curve of Archytas, the intersection of a torus and a cylinder,brings to mind a combination of this kind. According to Politics (307a onwards),Plato found the problem of weaving together temperaments to be isomorphic to theproblem of harmonizing low and high tones. And we know that musicians still usetwo medieties to fill intervals. Consequently, it is not impossible to suppose that Platoknew of the beautiful Archytas curve and that he had assessed the full significance ofthis entry into a higher dimension to guarantee the stability of the social system.

2.5.3. The doubling of the cube – going beyond Archytas: the evolutionof mathematical methods

While searching for solutions from a geometric point of view, the Greeksdeveloped a special technique that they called “analysis”. This consisted of assumingthat the problem was resolved, and then, through the study of the properties of thesolution, working backwards toward a problem equivalent to the given problem andwhich could be decided on the same principles.

Thus, to obtain the correct solution to the original problem, geometricians reversedthe procedure. First of all, the data are used to resolve the equivalent problem, derivedin the analysis. The original problem is then resolved using this solution. This secondprocedure is called “synthesis”. Euclid did not employ these methods in Elements.They have existed in mathematical and even philosophical traditions, however, sincethe days of antiquity.

Thus, the solution proposed by Menaechmus (380?–320?), a disciple of Plato andEudoxus of Cnidus (who was, himself, the disciple of Archytas of Tarentum) is quitetypical of this genre. In the analytical part of the demonstration, the geometriciansupposes the numbers on x and y, these being the desired proportional means. It isthen demonstrated that these numbers are equivalent to the results of the intersection


of three curves whose construction is assumed to be known. The “synthesis” thusconsists of introducing the curves, finding their intersection and showing that thisprocedure resolves the problems. Let us examine this more precisely.

2.5.3.1. Menaechmus’ solution

Inheriting the legacy of developments in mathematics from the time of Archytas,Manaechmus found two simple and elegant solutions:

1) from Hippocrates’ formula [2.1], he derives the following three equations(which we have written using our current analytical symbols):

y =1

ax2 (formula derived from the two expressions on the left-hand side) [2.11]

x =1

2ay2 (formula derived from the two expressions on the right-hand side) [2.12]

x = a 3"2 (formula derived from the equation: x3 = 2a3) [2.13]

Formulas [2.11] and [2.12] are those of parabolas. Formula [2.13] represents thelength of the edge of the desired cube. In geometric terms, the curves are:

a) a parabola with vertex O, axis ON , and thus symmetrical with respect to thex axis;

b) a parabola with vertex O, axis OM , and thus symmetrical with respect to they axis.

These curves determine, by their intersection, the point P such that:

OA.ON = PN2 OB.OM = PM2

As PN = OM and PM = ON , it follows that:

OA

OM=

OM

ON=

ON

OB[2.14]

OM and ON are, therefore, the two proportional means sought between OA andOB;


A

M

B

N

P

O

b

a

y

x

Figure 2.6. Menaechmus’ first solution

2) let b = 2a; then, Hippocrate of Chios’ formula becomes formula [2.2] ofArchytas. Hence, we once again have three equations:

x2 = ay (permutations of the extremes and means of the two left-hand sideexpressions) [2.15]

y2 = bx (permutations of the extremes and means of the two right-hand sideexpressions) [2.16]

xy = ab (permutations of the extremes and means of thefirst and third expressions) [2.17]

Menaechmus proceeded as above. He assumed that the problem was resolved. Hisreasoning, transposed into modern language, was as below:

With AO(= a) and OB(= b) being the two given lines, placed at right angles toeach other, we can trace OM(= x) along BO and ON(= y) along MO. We thencomplete the rectangle OMPN . We will then arrive again at formula [2.14].

From this, we can deduce that OB.OM = ON2 = PM2 (y2 = bx), whichproves that P is on a parabola with the vertex O, axis OM and where OB is thesection below the vertex.


A

M

B

N

P

O

b

a

y

x

Figure 2.7. Menaechmus’ second solution

Similarly, we can also deduce that: AO.OB = OM.ON = PN.PM (in otherwords, ab = xy), which proves that P is on a hyperbola with center O, asymptotesOM and ON and such that the rectangle contained by the lines PN , PM and P ,and, respectively, parallel to one asymptote and secant to another, is equal to a givenrectangle AO.OB.

If we then trace these two curves in accordance with the data, we determine thepoint P by the intersection of the two curves and, once again, find ourselves havingarrived at formula [2.14].

2.5.3.2. A brief overview of the other solutionsMenaechmus’ solutions are obviously simpler than those proposed by Archytas.

However, it was not the same period and the knowledge of this problem progressedover time.

As we have already observed, the problem of duplicating a cube led Greekgeometry to consider lines, surfaces and volumes other than those which it hadstudied until then. Other solutions to this problem of doubling a cube would appearsubsequently and would allow the discovery of new curves. Nicomedes’ solution, forinstance, would introduce the conchoid and Diocles’ solution would introduce thecissoid, a curve that was later studied by Descartes. With these curves, Greekgeometry, which with Apollonius had taken the form of a theory of conics, wouldfinally arrive on the threshold of mathematical analysis.


2.6. The trisection of an angle

Another problem, which is related to that of the doubling of a cube through itssolutions, is that of trisecting an angle.

According to Heath [HEA 21, p. 235], the problem of trisecting an angle seems toreally follow from attempts to construct regular polygons beyond the pentagon. Thetrisection of an angle would have been necessary to construct a regular polygon withnine sides or whose sides numbered any multiple of nine. Moreover, constructing aregular seven-sided polygon required the use of the famous curve that was inventedby Hippias of Elis (the quadratrix), although other methods that would soon emergewould also be used by the Greeks.

According to Dutens [DUT 12, p. 164], Proclus [PRO 33, p. 31] attributed onesolution to this problem to the Platonic school. However, we do not find any explicittrace of this in the Dialogues, although dividing into three was a constant problemdiscussed by Plato from the question of the tripartition of the soul in the Republic,up to the different trichotomies in Laws (dividing an army and the creation of States(683d–685d), dividing the number of lands, dividing a city (804c), learning to read(810b), classes of athletes and division of space, etc.).

When the problem was “solid”, as in the case of the doubling of a cube, the AncientGreeks failed to solve it using “planar” methods, that is, by constructing using a scaleand compass. Let us, for now, examine the proposed solutions. Two geometriciansknown to Plato, Hippias d’Elis and then Dinostrate, had tackled the problem.

2.6.1. Bold mathematicians

As we know, Hippias of Elis, introduced by Plato as a “sophist”, was thoroughlyridiculed by him in the dialogues that bear his name (Hippias Mineur and HippiasMajeur). Using the rhetorical device that we call “ambiguous eulogy”, and deeplyscornful of his interlocutor, the philosopher talks about his chief discovery4. Moreover,Hippias was actually a geometrician. It was Proclus (412–485) who, when reverifyingthe truth of the character, attributes to him the paternity of the quadratrix, a mechanicalcurve that would precisely carry out the trisection of an angle [PRO 48, p. 272].

Hippias constructed his trisectrix (which would later be called “quadratix”) usingtwo mobile scales. One of these is horizontal and set into motion with a vertical

4 See Plato, Hippias Mineur. Socrates, incidentally, only questions Hippias on banalities forwhich, visibly, there are no answers, and reproaches him, implicitly, for making money with his“science”. According to Plato, this is something which he prides himself on doing.


descending movement. The other scale is initially vertical, but can be inclined bysliding it until it finally overlays the first ruler, as shown in Figure 2.8:

Figure 2.8. Hippias’ quadratrix and its generator mechanism

We can note that the angle in the center, formed by the abscissa AB and a pointon the quadratrix is always proportional to its ordinate defined by the line AD. Inconsequence, the division of a segment of AD in equal parts, constructible using ascale and compass, will also divide the central angles into three equal angles.

Let us consider, for example, the angle BAE ('90$) and the square ABCD. Weform the quadratrix inscribed in this by using the mechanism described above. The lineAE divides the quadratrix at the point G, and the parallel on the side AB, originatingfrom G, then intercepts the side AD of the square at the point F ; the segment AFis proportional to the arc BE or, more precisely, the ratio of the lengths AF/AD isequal to the ratio of the lengths BE/BD.

From the point A, Hippias then created three equal, aligned segments (AN,NM ,MO) and traced the line OF . From M and N , he drew parallel lines to this line,thereby dividing AF into three equal points. The parallels to AB, drawn from theends of these segments, P,Q, intersect the quadratrix at the points U and T , whichprecisely divide the given angle into three equal angles.

The property of the quadratrix is that it is not constructible using the scale andcompass, though it may be constructible “point by point”, such that only a countableset of points may be obtained from it, as Lebesgue would one day observe. Only theabove mechanism makes it possible to have it wholly and it must be had wholly so asto be able to resolve the problem of trisecting the angle, because the useful points ofintersection are, a priori, indeterminate.


Using a similar procedure, we can naturally divide any angle smaller than or equalto 90$ into as many parts as we wish and the construction can be applied also to thefilling in of an interval using any number of segments.

Figure 2.9. The division of an angle into any number of parts

As we have said, the second geometrician to explore the problem of trisecting anangle is Dinostrate (390?–320?). We know very little about him other than the fact that(according to Proclus [PRO 48, book 2, Chapter 4]) he was Menaechmus’ brother, hewas Plato’s student and that around 350 B.C. he is likely to have used Hippias’ curveto resolve the problem of the quadrature of a circle, hence the name “quadratrix”.

Almost a century later, according to Pappus [PAP 89, book IV, proposition 25],this task would also be accomplished by Nicomedes (280?–210?) and “some other,younger geometricians” who tackled this difficult question. Nicomedes, who seems tohave lived in the same time as Eratoshenes and before Apollonious of Perga, is alsothe inventor of another trisecting curve on which he was very proud: the conchoid.But this takes us too far away from the age of the philosophers.

2.6.2. Plato, the tripartition of the soul and self-propulsion

As we have seen, Plato (staying faithful to Pythagorean procedures for thedimidiation of unity) quite often used dichotomic processes. This is especially triedin that part of the dialectic that seeks to produce definitions of objects or activities.


Sometimes, however, philosophers must carry out ternary cutting as they cannotcarry out binary cutting anymore. For example, in Phaedo, a simple episode inPlatonic teaching seems to accord a certain “simplicity” to the soul. However, thissimplicity, as G. Rodis-Lewis once remarked, was already limited and included theidea of the certain harmony, which the Republic would go on to make more precise.

In Republic, the soul, while never ceasing to be one also appeared “tripartite”,which the commentators found to be a serious problem. They were forced to struggleto resolve this, a problem that Plato himself never really resolved apart from withPhaedra and the thesis of self-propulsion, which was also rather astounding, andwhich is found in Timaeus, where souls entered a cosmic cycle.

A solution of this kind is problematic. Indeed, on closer inspection, Plato had, sofar, never prioritized the doctrines of movement. His opposition to Heraclitus, inparticular, had caused him to refuse this. According to a text by Plutarch, whichshould be taken with some reservations, writes B. Vitrac, Plato would havedisapproved of the introduction of movement in geometry, which would constitute asort of “corruption” [VIT 05c]. At the same time, Vitrac rightly points out that inTimaeus, the real trajectory of heavenly bodies was compared to a curve and Platodescribed a “model with two homocentric spheres” and specified that “this implieshelicoidal (elika) trajectories for heavenly bodies subject to two movements of theSelf and the Other”5. He admits, moreover, that “apart from these kinds ofconsiderations, the geometrician could study and use these new objects by‘forgetting’ their physical origin and the context that justified taking them intoconsideration. The geometrician could, for example, generalize the spherical spiral,which takes into account the annual apparent movement of the sun, and introduceother types of curves resulting in the same manner (by the combination of two simplemovements) such as the cylindrical helix or the spiral on the lateral surface of a cone,etc. They can also accept that geometric figures are put into movement to allow themto resolve difficult geometric problems such as that of the intercalation of two linesthat are proportional means between two given lines”6.

However, before these solutions were applied to the cosmology that includes, aswe have seen, Archytas’ solution to the problem of doubling a cube, it is possible thatthey were able to inform the question of the human soul and its internal harmonization.

How does Plato conceive of harmony between the elements of the soul (reason,heart, desire) in Book IV of the Republic? Like that of the harmony in the citydescribed in Book III, of course. But what else? In 422e, it is said that this harmony

5 See Plato, Hippias Minor, p. 8.6 See Plato, Hippias Minor, p. 8.


must be thought of as a perfect accord comparable to musical accord between Nete,Hypate and Mese and any intermediaries that may exist. Moreover, if theinstrumental model, as used in Phaedon earlier (referred to by Simmias), is the lyre,which is a heptachord, it is absolutely essential that these intermediaries exist. Evenif Plato here says nothing on this subject, we know that at a later period, around thetime of Timaeus, the Greek musical range would indeed be the paradigm in which theindividual soul – like the soul of the world, of which this would be an extension –was constructed. The Greek system contains seven intervals and eight notes.According to expert interpreters, the general 2 : 1 interval, called dia pasôn, isarticulated across two medieties, one harmonic and one arithmetic. This gives theseries:

1, 1 +1

3, 1 +

1

2, 2

the interval:

[1 +1

3, 1 +

1

2]

which in turn, may be filled in the same manner by two mediants of the same type. Wethen arrive at the series: Nete, paranete, trite, paramese, Mese, likhanos, parhypate,Hypate. The division here goes well beyond that of the division referred to with respectto the doubling of a cube and, moreover, we must explain how the soul, which is one,can pass continuously from one “tonality” to another.

Plato explains nothing and the musical model, in reality, remains shaky as themusical intervals are unequal, while the parts of the soul, in the case of awell-tempered soul, must be in harmony with each other without any imbalancebetween them. This is so true that the “heart” must side with reason in order tomaster the desires.

It could, therefore, be possible that the tripartition of the soul is like the trisection ofan angle proposed by Hippias, carried out by the quadratrix that allows any subdivisionof a segment into equal subsegments.

2.6.3. A very essential shell

In Phaedra, in 250c, we see that our fleshy envelope “this sepulchre that we carrywith us and that we call our body, binds us to itself like the oyster to its shell (ostreoutropon)”. The is a free translation of the text. The literal translation would be “in themanner of the oyster”. But it conveys the meaning.

It is quite ironic that Nicomedes, a mathematician of the 2nd century B.C., is theinventor of a curve that would have astonished Plato: the conchoid (from the Latin


concha, shell). This is obtained by starting from a fixed point O, another curve, anda certain distance, d. There are thus conchoids possible from a circle, an ellipse andany other sort of plane curve. The simplest conchoid, however, is undoubtedly theconchoid of a straight line. This is what Nicomedes invented. Like Hippias, he madeuse of a mechanical means of doing this. Today, it is defined by the polar equation:

# =a

cos$+ d

where a is the distance from the pole O to the directrix.

yP

NM

H

Figure 2.10. The conchoids of Nicomedes. For a color version of thisfigure, see www.iste.co.uk/parrochia/philosophy.zip

The conchoids of Nicomedes are also trisectrixes, just like the quadratrix ofHippias. A different conchoid corresponds to every angle ! to be trisected.

To carry out the trisection (see Figure 2.10), we construct a right triangle OHI ,with the right angle at H , such that the angle to trisect, !, is !OIH . We then constructthe conchoid for the line (IH) from the pole O and the modulus OI .


We then have with !IOH = ! : a = OH and d = OI =a

cos!. The equation for

the conchoid is therefore:

# =a

cos$+

a

cos!

The intersection of the curve with the circle that has the center I , passing throughO, makes it possible to determine two points M and N . And because of the propertiesof the conchoid, we can demonstrate that the angle !NIP trisects the angle !OIH , oragain, that the angle !NIP is a third of the angle !OIH .

Thus, the conchoid was reclaimed – at least its mathematical form –as it could be used to carry out trisections. A modern philosopher may even go so faras to say that the body allows structuring of the soul, something that a Platonicthinker would be unable to contest.

2.6.4. A final excercus

The bisection or trisection of an angle will, of course, lead to the appearance oflines (these are, respectively, the bisectrix and the trisectrix). In the case of a triangle,these lines must intersect each other. In the first case, we know that the bisectricesof a triangle have a unique point of concurrence, the orthocenter of the triangle or thecenter of the inscribed circle. In the case of the trisection, the study of the intersectionsof the trisectrices could not be carried out earlier than the late 19th Century. Indeed,mathematicians had to wait until 1898 for the British mathematician Frank Morley(1860–1937) to propound a remarkable theorem:

THEOREM.– [Morley, 1898] The intersections of the trisectrices of the angles of anytriangle form an equilateral triangle.

This result, both beautiful and astonishing, today allows for differentdemonstrations (geometric, trigonometric, using complexes, or even in grouptheory). It shows how the introduction of symmetry (the trisectrices) in an initiallyasymmetric situation (any triangle) produces internal symmetry (the equilateraltriangle), as though a little order in disorder leads to stability. This would clearly haverejoiced Plato’s heart. He would have seen, in this hidden equilateral triangle lodgedinside every kind of triangle, a precious and new image of the soul.

2.7. Impossible problems and badly formulated problems

The demonstrations of the impossibility of geometric solutions for the problem oftrisecting an angle or duplication of a cube (or, to be more precise, the demonstration


of the non-constructibility of scale- and-compass solutions) were carried out byPierre-Laurent Wantzel (1814–1848) in 1837. It must be pointed out, however, thatfrom 1800 or so onwards, Gauss had got wind of these questions. In 1882,Lindemann demonstrated the impossibility of the last problem posed by Greekgeometry, that of squaring a circle (which we will approach soon).

Figure 2.11. Morley’s theorem. For a color version of this figure, seewww.iste.co.uk/parrochia/philosophy.zip

The final reason why the problem of doubling a cube cannot be resolved byclassical geometry and its scale-and-compass constructions, is related to the conceptsof the Field and Galois theory, which began to be developed only in the 19th Century.

Let us briefly review these concepts.

2.8. The modern demonstration

To find a criterion for the constructibility of numbers, it is convenient to placeourselves in the set of complex numbers, as these correspond to a place and numbersthat can be constructed using a scale and compass are those in plane geometry.

Thus, let z1, z2, . . . , zn ( C and F = Q(z1, z2, . . . , zn). A number z is said tobe constructible, from z1, z2, . . . , zn, if z is contained in a subbody of C of the shapeF (u1, u2, . . . , un), where u2

i ( F and every u2i ( F (u1, u2, . . . , un).

Starting from a set of constructible points S = {0, 1}, we can also recursivelydefine the set C of constructible numbers as being the set C containing Q and such


that, if p(x) is a quadratic polynomial with coefficients in C, then its roots (whetherreal or not) are also in C.

THEOREM.– [Theoreom of impossibility] The Delian constant 3"2 is not a

constructible number.

The shortened proof. In effect, [Q( 3"2): Q] = 3 is not a power of 2 as x3 # 2 is

irreducible over Q [JAC 85, p. 221].

This being the case, it is possible to reflect on the history of the problems and theirmovement: the doubling of a cube, like the problem of trisecting an angle or that ofsquaring a circle, is not impossible in itself. These are only impossible using specificmethods that science could authorize in a thorough manner in a given historical period.

As Bachelard shows, such problems are not, therefore, absolute borders: they onlyreveal the limits of a certain kind of knowledge. This is de facto always the case withknowledge, which is constantly subject to historical evolutions. Impossible problems,therefore, are only problems that are wrongly posed, as the following text shows:

“Scientifically, the border of knowledge appears to mark only amomentary arrest in thought. It would be difficult to trace objectively. Itseems that the limitation of scientific thought is desirable whether it is interms of the program or of the absolute obstacle, in terms of possibilityrather than impossibility [...].

Philosophically, every absolute border proposed in science is a mark of aproblem badly posed. It is impossible to richly think of an impossibility.As soon as an epistemological border appears clear, it is then that itassumes the right to sever itself from the subject of first intuitions. Butfirst intuitions are always intuitions to be rectified. When a method ofscientific research loses its fertility, it is that the point of departure is toointuitive, too schematic; it is that the base of organization is too limited.The duty of scientific philosophy seems then very clear. It is necessary towear away all parts of initial limitations, to reform non-scientificknowledge that still fetters scientific knowledge” [BAC 70, pp. 84–85].

The border of knowledge is thus in no way comparable to a Jordan curve, whichtraces the absolute limit between what we know and what we do not know. It is bothfluid and fluctuating, like the border that G. Devereux [DEV 72, pp. 51–63] seems toassign to the division between what lies within and outside of a subject. In otherwords, a series of divisions analogous to a Dedekind curve, mobile as it is created denovo at each instant by events that are produced on it and make it up. For Bachelard,it is on this fluctuating border that we rectify our initial intuitions, which are alwaystoo simple and too rigid. This border is redrawn as a result of new research


programmes and new methods of investigation. One of the most useful tasks of aphilosopher would be to precisely help knowledge overcome and transgress theseprovisional boundaries, which are only momentary impossibilities, signs of badlyposed problems7, in order to freely allow new ways of posing the problem and newmethods of resolution. The squaring of a circle would one day be possible because ofintegral calculation. The doubling of a square and trisection of an angle would nothold out long against analytical geometry. On the other hand, a timid interpretation ofthe precaution principle risks paralyzing this spirit of innovation, which approachesthe borders of knowledge and shoots past, penetrating unexplored, virgin territoriesof the imagination. If we do not explore these new territories, we have no chance ofmaking any progress.

7 This proposition must, however, be accompanied by a few qualifiers. It so happens that inphysics one is able to very precisely define what a “well-formulated” problem is. Hadamard hascharacterized this kind of problem in a very precise manner: the solution exists, it is unique andit depends in a continuous manner on data, within the framework of a reasonable topology. Allproblems, however, are not of this nature and for some the solution varies considerably for asmall change in data. In this case also, however, physicists have developed appropriate methods:see [TIK 76].

3

Quadratures, Trigonometryand Transcendance

The knowledge of irrational numbers would grow until the 17th Century and evensome time beyond that. One of the earliest discoveries, however, was that irrationalsas a class consisted of different types of numbers, of which some were never thesolution to an algebraic equation. One of these numbers (which would someday becalled “transcendants”) was the subject of much detailed study from the earliest daysof antiquity.

In effect, with the appearance of the form of the circle, a shape both fascinatingand enigmatic, ancient Greece was faced with a problem. Was it possible to comparethe area of a circle to the area of a polygon (especially that of the polygon that seemedclosest to it, the square)? And if so, how? This was the problem of squaring a circle,i.e. how to use a scale and compass to construct a square having the same area as thatof a circle with a given area. Assuming that the radius of the circle is R = 1, whichgives it an area of S = "R2 = ", the problem is then that of finding a number a, suchthat:

a2 = "

Different methods would be used to approach this problem. One of the notableapproaches was the “method of exhaustion” used by Archimedes (3rd Century B.C.),who tried to approximate the perimeter of the circle by excess or defect and throughfalse assertions that must be proven false, using regular polygons with an increasingnumber of sides, even substituting it with an indefinitely broken line where eachsegment (growing shorter and shorter) will envelope its curvature as best possible.Contrary to appearances, this is not the forerunner to the infinitesimal calculation,which we shall discuss later.



3.1. " – the mysterious number

Before thinking of squaring a circle, it would be useful to define ". Determiningthe number " = 3, 1415926... is not, however, trivial and even poses a slightlyparadoxical problem. Today, we calculate perimeters and areas of circles using " andwe write formulas such as P = 2"R, S = "R2, where P is the perimeter and S isthe surface area.

The question then is: how did we discover " and calculate its value?

This problem was posed early on in human history.

Historically, the existence of a constant ratio between the circumference anddiameter of a circle of any dimensions was first observed through a number ofempirical practices.

The first approach undoubtedly consisted of determining the number " usingpurely physical measurement. This is how we obtained the approximate and highlyrough values, such as we find in the Bible (Book of Kings, 7:23). The Biblicalexample is of a circle of 30 cubits whose diameter is 10 cubits. This gives " = 3, avalue that cannot be said to be very precise.

A surprisingly good approximation of " can also be found in the famous Rhindpapyrus (Egypt), around 1900 B.C. It indicates that we obtain the area, A, of a circleas a function of its diameter d using the formula:

"R2 = "d2

4= (d# d

9)2 hence: " =

4(d# d9 )

2

d2

If we take d = 1, it follows:

" = 4(1# 1

9)2 = 4(

8

9)2 = 22(

8

9)2 = (

16

9)2 = 3.16

An even better approximation was discovered toward the end of the 5th Centuryby the Chinese mathematician Tsu Ch’ung-Chih, with " ) 355

113 .

We do not, however, know how these approximations were obtained.

At this stage, which is still empiricial, nothing indicates that " is a calculablenumber, that is, that a method exists which makes it possible to determine with anarbitrarily large precision. Before demonstrating which algorithm is used to finallyarrive at this, we will restrict ourselves to a few words on the false problem of thesquaring of the circle.

Quadratures, Trigonometry and Transcendance 53

3.2. The error of the “squarers”

The problem of exactly computing the area of the circle clearly fascinated theGreeks.

The Greek writer Plutarch reports that the philosopher Anaxagoras (who wasimprisoned in Athens for “impiety”) is likely to have invented this enigma ofsquaring the circle around 430 B.C. [MON 31, p. 34]. This problem finds mention,above all, in the later commentaries on Aristotle’s works. For this reason, the order inwhich the ideas appeared is quite uncertain. The most considerable works of the 5thCentury B.C. were those of Hippocrates of Chios, Antiphon, Bryson of Heraclea andHippias of Elis.

It appears that, much before Archimedes, it was Antiphon who first had the ideaof approximating the perimeter of the circle using the perimeter of regular inscribedpolygons with an increasing number of sides [BRU 93, p. 156]. Bryson of Heracleaperfected this procedure by framing a circle using both inscribed and circumscribedpolygons, retaining the mean value each time [REY 46, pp. 224–227]. As concernsHippias of Elis, around 425 B.C. he conceived of dividing any given angle into threeusing an auxiliary curve, described by composing a uniform rectilinear movement witha uniform circular movement. One century later, Dinostrates would discover that it waspossible to use this same curve – the “quadratrix” – to construct a segment of length2! and, from here, use various elementary geometric transformations to construct asquare with an area of ". However, this kind of a curve (which would one day becalled “transcendent”) was impossible to construct using a scale and compass and,thus, this solution could not be considered a “geometric” one [DEL 97, p. 54].

Toward 440 B.C., Hippocrates of Chios studied this problem methodically andrecognized the impossibility of directly squaring the circle. He did believe,nonetheless, that it was possible to avoid this obstacle by first squaring what is calledthe “lune”. This problem, according to Eudemus, was resolved as per a methodreported by Simplicius in his commentary on Aristotle’s physics [SIM 85,pp. 56–57].

As we can see in Figure 3.1, the area, SA!, of the semicircle with diameter A" isequal to the surface of the quadrant A"!.

In effect:

A"2 = A!2 +!"2 = 2A!2 = 2(AB

2)2 = 2

AB2

4=

AB2

2

We consequently have:

SA! = "(A"

2)2 =

"A"2

8= "

AB2

2

8= "

AB2

16= "(

AB

4)2


Figure 3.1. Squaring the lune

As the area of the circle AB is "(AB2 )2 = "AB2

4 , the area of the semicircle A" isnone other than the area of the semicircle AB, divided by 4 (Euclid, XII, 2)1. But thequadrant A"! and the semicircle have a common region, the space between the lineA" and the arc of the circle with centre ! that it underlies. If we redivide this space,we arrive at the equality of the respective areas of the lune AE" and the right triangleAD". Moreover, the area of this triangle is a quarter of that of the square with sideA" and is then itself equal to a square of side A!

2 .

The squarers grew optimistic at the thought that the area of a lune, that is, asurface between two curves, could be likened to that of a polygon. Alas! They weresoon disillusioned. As Abel Rey pointed out2, not every lune is squareable and thecircle, consequently, is not covered by this method, despite the repeated efforts of a

1 We thus verify that the circles (or semicircles) are related as the squares of their diameters.2 Around the middle of the 5th Century, Hippocrates had succeeded in squaring three differentlunes. But these were three particular and special types of lunes. For a long time, it was evenbelieved that only these were squareable and could be constructed using scale and compass.Hankel still believed it. In reality, M.J. Wallen of Abo, in 1776, and Clausen, in 1840, perceivedthat the squaring of two other lunes was possible. Nonetheless, these five squareable lunes areexceptional cases. In general, the lune is no more squareable than the circle [REY 48].


“formidable army of squarers” [KRO 05, p. 81] who all, one after the other, fell forthis fallacy for the better part of a millenium3.

3.3. The explicit computation of "

It was Archimedes (287–212 B.C.) who first proposed an algorithm to compute ",that is, a series of steps that could yield a result in finite time.

In a famous treatise that has survived till today [ARC 70a], the physicist managesto demonstrate the following three theorems:

1) the area of a circle is equal to the area of the right triangle where one of thesides of the right angle is the radius and the other the perimeter of the circle. The areaof a circle of radius R and of perimeter P , is then: R%P

2 ;

2) the ratio between the area of the circle and the square of its diameter, D = 2Ris about 11

14 ;

3) the ratio PD of the perimeter of the circle to its diameter is between 3 + 10

71 and3 + 10

70 .

The first theorem resolves the squaring and likens it to the construction of asegment of length ". The third theorem yields a value that is both simple ( 227 ) andprecise enough for current applications. The second theorem is a corollary to theother two. The rest, already known from Euclid and Archimedes, stops withspecifying the proportionality coefficient.

To demonstrate these three theorems, Archimedes went back to Bryson’s concept:approximating the circle using inscribed and circumscribed polygons by multiplyingthe points of contact.

We must note that Archimedes’ method yielded a remarkable manner ofapproaching ":

1) it can be seen that it is always possible to inscribe a regular polygon in a circle;

2) we begin with the simplest of the regular polygons, the equilateral triangle;

3) the number of sides of the triangle is systematically doubled. In the firstiteration, the initial equilateral triangle (three sides) is transformed into a hexagon(six sides);

3 In France, the Academy of Sciences finally warned that they would henceforth refuse toexamine the so-called “demonstrations” that were regularly submitted before them and whichmust have undoubtedly arrived before them in the same numbers as the perpetual motion case.


4) it is enough to continue this process and double the number of sides on eachiteration, until the point where the polygon with an infinity of sides that is obtained ifthis operation is carried out indefinitely. This polygon will be the same as the circle.

The perimeter of the circle being P = "D = 2"R, it is clear that if we take acircle of R = 1

2 , then P = ", and the perimeter of the polygon inscribed in the circleis thus a good approximation of this number.

Figure 3.2. Computing "

Let Pn thus be the perimeter of a regular polygon of n sides in a circle of R = 12 .

The length of a side is Pnn and Archimedes thus observes (translated into today’s

notations):

" = limn"#

1

nPn

It is clear that we may easily discover the value of P6 – which is the case witha hexagon. In this case, the circle is effectively divided into six equilateral triangles,such that the sides of the hexagon each have the value of the radius, that is 1

2 . Fromthis, it results that the circumference of the polygon is: 6! 1

2 = 3.

We can then evaluate the limit using the series: P6, P12, P24, P48... obtained bysuccessively doubling the number of sides of a polygon. This procedure is knownthen by the name: “method of exhaustion”.

The question is: knowing un = 1nPn the length of a side of an n-gon, or polygone

of n sides, how do we arrive at the value of u2n = 12nP2n?

To resolve this question, we must turn to trigonometry.


3.4. Trigonometric considerations

The Greeks were undoubtedly the first to discover that the relation between thesides and the angles of a right triangle could be expressed using constant ratios calledthe sine, cosine, tangent and cotangent4. The study of these ratios, called trigonometry,is useful in both computing the polygonal areas (any polygon can be reduced to a sumof triangles) and astronomical distances (where the combination of angular measuresand distances that are already known makes it possible to then calculate others). It canbe immediately observed that for a circle with a unit radius, the two sides of the rightangle are themselves the sine and cosine of the central angle, $, such that, using thePythagoras theorem, we must have:

cos2$ + sin2$ = 1

Hence:

sin2$ = 1# cos2$ = (1 + cos$ )(1# cos$ ) [3.1]

But we also demonstrate that:

cos2$

2=

1 + cos$

2[3.2]

Upon substituting [3.2] in [3.1], we obtain:

sin2$ = (2 cos2$

2)(1# cos$ ) = (2 cos2

$

2)(1#

"cos2$)

Hence:

sin2$ = 2(1# sin2 $

2)(1# (

!1# sin2$)) [3.3]

Moreover, we have:

sin$ = 2 sin$

2cos

$

2

4 The “versed-sine” or versine function was added to this later.


Hence, on squaring:

sin2$ = 4 sin2 $

2cos2

$

2= 4 sin2 $

2(1# sin2 $

2) [3.4]

On comparing [3.3] and [3.4], we obtain:

4 sin2 $

2= 2(1#

!1# sin2$)

That is:

sin2 $

2= 1/2(1#

!1# sin2)

And, finally:

sin$

2=

"1/2(1#

!1# sin2$)

We thus posit:

sin$

2= u2n sin$ = un

Hence the recurrence relation:

u2n =

#1

2(1#

!1# u2

n) [3.5]

We know that we have u6 = 12 , which leads to P6 = 3.

We can, thus, easily calculate the successive approximations of ".

The values are defined as follows:

u2n =1

2nP2n =& P2n = 2nu2n

and we obtain Table 3.1.

In the eight iteration, we see that we have five digits after the decimal points andthat " then approaches 10&4. This algorithm, which converges slowly, was laterreplaced by more efficient algorithms, which converged much more rapidly.


u12 = 0.258819045 P12 = 3.105828541u24 = 0.130526192 P24 = 3.132628613u48 = 0.065403129 P48 = 3.139350203u96 = 0.032719083 P96 = 3.141031951u192 = 0.016361732 P192 = 3.14145247u384 = 0.00818114 P384 = 3.141557603u768 = 0.004090604 P768 = 3.141583864u1356 = 0.002045306 P1536 = 3.141590341

.... ...

Table 3.1. The first approximations of "

3.5. The paradoxical philosophy of Nicholas of Cusa

One could think that the computation methods used in Archimedes’ work wouldput an end to the speculations of the squarers. However, they did nothing toward thisend. Even as late as the 16th Century, philosophers like Nicholas of Cusa would workon the question of squaring a circle. The only originality of his research was that ithad several important consequences for philosophy.

3.5.1. An attempt at computing an approximate value for "

The author’s first treatise on this question was titled: De transformationibusgeometricis (Geometric Transformations). As Morritz Cantor [CAN 00,pp. 194–195] points out, it bears the dedication: “A Paulum magistri dominiciPhysicum Florentinum”, that is, “To the doctor of Florence Paulus” (a certain PaoloToscanelli), who we know was the author’s classmate at Padua. The author of thededication declared that the book dealt with the transformations of curves to linesand lines to curves. As a rational ratio is not possible between the two, the secretmust lie in an exact coincidence of their extremities. Simply put, this coincidence,which can be applied to the maximum (the polygon that approaches an unknowncircle), must be sought from the minimum (the smallest polygon is the triangle).

Nicholas of Cusa seems to reason from the number of angles in the figures: thesmallest of these is composed of three angles, the largest contains an infinite number.Let us use one figure to explain his reasoning: let BCD be a given triangle and AF bethe radius of the inscribed circle (always the smallest). The radius of the outside circle,which may circumscribe any regular polygon, is evidently larger than the triangle. Theradius of the desired circle, whose perimeter coincides with that of the triangle, musttherefore be greater than AF and smaller than AB.

Let us now divide FB into four equal parts. Let I, E, L be points connected toA by the lines AI,AE and AL, which are themselves extended on the other side to


IK, EH, LM , such that the ratio between AK/AI,AH/AE and AM/AL is thesame as, respectively, BF/IF,BI/EI and BL/LE. It is clear that we then have:

IK =AI

8

Figure 3.3. The coincidence of extremities accordingto Nicholas of Cusa

Indeed:

BC

IF=

AI

IKhence: IK =

(AI.IF )

BC

But:

IF =BF

4=

BC

8

Hence:

IK =AI.(BC

8 )

BC=

AI

8

Moreover, given that according to the hypothesis:

BC

FE=

AE

EHand

BC

FL=

AL

LM


we obtain:

EH =AE

4and LM = 3

AL

8

Let us now take I, not far from F, and M, not far from B. The rapprochement ofthese points is so evident, declares Nicholas of Cusa, that it must inspire wonder, justlike the following result (slightly inadequate, to tell the truth).

Let BC = 8, which gives 8 x 3 = 24 as the perimeter of the equilateral triangle.Assuming that we are able to coincide the triangle and the circle, this would signifythat we have 24 = 24"AH , or " = 12/AH . We also have the following equalities:

AF = AB/2, BF = 4, 3AF2 = 16, AF = 4/"3, AE2 = 16/3 + 4 = 28/3

AE ="84/3, AH = 5/4AL = (5/12)

"84

And finally:

" = 144/5"84 =

"9.87428571428571... = 3.142337

the value lying between 3 + 1/7 (= 3.142857...) and 3 + 10/71 (= 3.140845...), andwhich is more precise than the first rational approximation discovered by Archimedes.

3.5.2. Philosophical extension

This possibility of bringing together, as far as possible, things that are as far apartas a portion of a curve and a portion of a circle suggested to Nicholas of Cusa(1401–1467) (who was, himself, the author of a number of attempts at quadrature)that the essence of reality, God, is, in fact, an entity that contains within themselvesall the contradictions in the universe. That God is himself (or herself!) coincidentiaoppositorum, the coincidence of the contradictions or extremes5. Nicholas of Cusawas thus close to conceiving what the 17th Century mathematician Girard Desargueswould begin to discover and what 19th Century geometry (with Monge and Poncelet)would truly, rigorously theorize: namely, the projective properties of shapes and theidentification, to infinity, of shapes that in the finite are distinct. Here is whatNicholas of Cusa, at any rate, wrote in his famous work, De la docte ignorance, withsupporting diagram [CUE 30].

5 It is also possible that this idea is logically translated by overlaying the sides of the notorious“square of opposition”, co-invented by Apuleius and Aristotle. For more on this, see [PAR 15a].


“Operating in this way, then, and beginning under the guidance of the maximumTruth, I affirm what the holy men and the most exalted intellects who appliedthemselves to figures have stated in various ways. The most devout Anselmcompared the maximum Truth to infinite rectitude. (Let me, following him, haverecourse to the figure of rectitude, which I picture as a straight line.) Others who arevery talented compared, to the Super-blessed Trinity, a triangle consisting of threeequal right angles. Since, necessarily, such a triangle has infinite sides, as will beshown, it can be called an infinite triangle. (These men I will also follow.) Otherswho have attempted to figure infinite oneness have spoken of God as an infinitecircle. But those who considered the most actual existence of God affirmed that He isan infinite sphere, as it were. I will show that all of these [men] have rightlyconceived of the Maximum and that the opinion of them all is a single opinion”[CUE 30, pp. 60–62].

Two observations can be made about this text. On the one hand, through thedifferent metaphors that are used to refer to them, it seems that God, or the AbsoluteTruth, was successively likened to an infinite line, an infinite triangle, an infinitecircle and an infinite sphere, thus rendering all these figures equivalent. On the otherhand, it was not until Girard Desargues, and above all Monge and Poncelet, that theequivalence, redefined as projective equivalence, finds a real legitimacy inmathematical reasoning. In brief, Nicholas of Cusa’s intuitions had to be seriouslyrectified to become rationally acceptable. The next part of his text, however (section13), draws the logical conclusion from his earlier remarks.

“I maintain, therefore, that if there were an infinite line, it would be a straightline, a triangle, a circle, and a sphere. And likewise if there were an infinite sphere, itwould be a circle, a triangle, and a line. And the same thing must be said about aninfinite triangle and an infinite circle. First of all, it is evident that an infinite linewould be a straight line: The diameter of a circle is a straight line, and thecircumference is a curved line which is greater than the diameter. So if the curvedline becomes less curved in proportion to the increased circumference of the circle,then the circumference of the maximum circle, which cannot be greater, is minimallycurved and therefore maximally straight. Hence, the minimum coincides with themaximum? To such an extent that we can visually recognize that it is necessary forthe maximum line to be maximally straight and minimally curved. Not even a scrupleof doubt about this can remain when we see in the figure here at the side that arc CDof the larger circle is less curved than arc EF of the smaller circle, and that arc EF isless curved than arc GH of the still smaller circle. Hence, the straight line AB will bethe arc of the maximum circle, which cannot be greater. And thus we see that amaximum, infinite line is, necessarily, the straightest; and to it no curvature isopposed. Indeed, in the maximum line curvature is straightness. And this is the firstthing [which was] to be proved. Secondly, I said that an infinite line is a maximumtriangle, a maximum circle, and a [maximum] sphere. In order to demonstrate this,we must in the case of finite lines see what is present in the potency of a finite line.


And that which we are examining will become clearer to us on the basis of the factthat an infinite line is, actually, whatever is present in the potency of a finite line. Tobegin with, we know that a line finite in length can be longer and straighter; and Ihave just proved that the maximum line is the longest and straightest”.

a b

d

f

h

r

Figure 3.4. The “passions” of the maximal and infinite line

Here, identifying the line and the circle (at infinity) reflects on the finite definitionof these entities: thus, a finite line is more or less straight (or, respectively, more orless curved), thus making it possible to, at some limit, liken the curve to the line. Butthe infinitesimal calculation, which is the only one that can support this identificationwas still some way away. In Nicholas of Cusa’s mathematical trials, there was stillonly a focus on resolving the problem of squaring a circle.

At this time, as we know, there was still a long, long way to go before the exactnature of the number " could be known.

3.6. What came next and the conclusion to the history of "

Using his recurrence relation, Archimedes could, in principle, have calculated anyapproximation of ". He stopped, as we have seen, at a rough approximation and thenlost interest in the problem of the intrinsic reality of this number. His procedure ofinfinite approximation did, however, lead one to suspect that the number was notrational.

In the Classical Age, many authors including Viète, Descartes, Leibniz and Eulerproposed formulas to represent ", its multiples and powers. As François Le Lionnaisobserved in his book on Remarkable Numbers (Les Nombres remarquables), thechief motivations behind this research was always the desire to resolve the problemof squaring a circle, first using geometry and then analysis [LE 83, p. 50].

" in this context led to the introduction of new mathematical expressions that werecompletely novel and that no one had dared write until then.


3.6.1. The age of infinite products

The philosopher Merleau-Ponty was able to define the grand rationalism of theClassical Age as an “innocent way of thinking from the infinite” [MER 60, p. 188]. Asthe philosophers of this time had already started lagging behind the mathematicians,it can be considered that it had already started by the end of the 16th Century, whenwe saw the first “infinite products” appear. Thus, Viète, proposed, in 1579:

2

"=

#1

2.(

#1

2+

1

2

#1

2).((

#1

2+

1

2

#1

2) +

1

2...)

Wallis wrote, in 1655, that:

"

2=

2.2

1.3.4.4.

3.5.6.6

5.7....

2n.2n

(2n# 1).(2n# 1)

In 1671, the English mathematician James Gregory gave the world the classicseries, rediscovered in 1674 by Leibniz:

"

4= 1# 1

3+

1

5# 1

7+ ...

The problem with these texts is that they are unusable in practice as theconvergance is too slow. In addition, they do not explain anything and, by the end ofthe 17th Century, Huyghens would remark to Wallis that people still did not knowwhether " was rational or not.

3.6.2. Machin’s algorithm

In the 18th Century, a very different type of algorithm was proposed by JohnMachin (1680–1752). It was based on the formula:

" = 16 Arctg1

5# 4 Arctg

1

239

which can be verified using trigonometric addition formulas. The function Arctg x canbe represented by the series:

Arctg x =x

1# x3

3+

x5

5# ...

Machin’s algorithm consisted of computing the successive approximations ("m)of " given by:

"m = 16 Arctgm1

5# 4 Arctgm

1

239


where:

Arctgm x =m$

k=0

(#1)x2k+1

2k + 1

This remarkable algorithm converges much more rapidly than Archimedes’algorithm, discussed earlier. Moreover, the computation here does not require theextraction of the square roots. As summarized in the formulas above, it could beeasily specified in the form of a computer program. In any case, it allowed JohnMachin to determine the first hundred decimals of ".

3.6.3. The problem of the nature of "

What type of number is "? It was immediately seen that it is not a whole numberand the suspicion rapidly arose that it is not a rational number. However, antiquitycontented itself with declaring (but not demonstrating), with Aristotle, that thecircumference and diameter of a circle were incommensurable.

Thinkers and mathematicians had to wait until the second half of the 18th Centuryand the work of Johann Heinrich Lambert (1728–1777) for the first demonstration ofthe irrationality of ". Lambert gave this demonstration in 1766 in his work on thequestion of squaring a circle [LAM 08, pp. 194–212]. This demonstration, foundedon continuous fractions, began from the continuous, infinite fraction:

tg z =z

1# z2

3& z2

5" z27"...

[3.6]

From this, Lambert deduces the irrationality of tgz for all the rational argumentsz $=0, as this development as a continuous fraction is typical of an irrational number.

Moreover, as it is known that tg !4 = 1, it follows that !

4 is strongly irrationaland, therefore, " is irrational. In effect, an irrational number may be approached byan infinite series of rational numbers. However, in order for 1, which is rational andeven whole, to be able to give way to an approximation using continuous fractions, itis essential that the numbers z, which approach it, be irrational. However, Lambert’sdemonstration was incomplete as it lacked a lemma on the irrationality of certaincontinuous fractions that had a particularly rapid convergence.

The next step was taken by the French mathematician Adrien-Marie Legendre(1752–1833). In the sixth edition of his work, Eléments de Géométrie (Elements ofGeometry), note IV, Legendre demonstrated that for any positive, rational number


q, Lambert’s continuous fraction [3.6] is irrational for "q tg"q. Consequently, we

cannot have " ="q with q as a positive rational, as Lambert’s continuous fraction is

no longer irrational. Thus, " cannot be the root of a rational number. And if it cannotbe the root of a rational number, it has to be irrational.

3.6.4. Numerical and philosophical transcendance: Kant, Lambert andLegendre

In the conclusion to his article on the irrationality of ", Legendre conjectured that" must be a number that is even more singular than the ordinary irrationals. Here iswhat he wrote in 1806:

“It is probable that " is not even contained in the algebraic irrational l. In otherwords, it cannot be the root of an algebraic equation with a finite number of terms andirrational coefficients. This theorem, however, seems difficult to rigorously prove”.

Legendre, thus, put forth the particularly hardy conjecture that ", in some way,exceeded “ordinary” irrationals in irrationality. He decided, therefore, to call numberssuch as ", that were not algebraic (that is, they had no solution equations with a finitenumber of terms and rational coefficients) transcendants. This was in the sense thatthey “went beyond” or “transcended” all rationality – even that of continuous fractions(omnem rationem transcendunt).

No philosopher seems to have noticed, until this time, that transposed into naturallanguage and applied to Plato’s dichotomous processes, such a statement wouldsignify that there may exist, in a language, terms that would be absolutely out ofreach of these approximations using successive divisions. Or, in more mathematicalterms, out of reach of the filters or ultrafilters6 that are used to try and arrive atprecise definitions using the genus-differentia definition. These methods were usedby Aristotle as well as in Platonic dialogue and, more generally, in any linguisticprocess of decomposition, such as those used by specialists in componential analysisin semantics. This was then the recognition that certain problematic terms inlanguage (and thus, certain types of realities) could be indeterminable or, at any rate,out of reach of the processes of progressive approximation, even infinite. What theseterms could be is, evidently, an open question. But the most problematic ofphilosophical problems are certainly involved.

6 According to a suggestion made by G.-G. Granger, it is possible to show that this was indeedthe mathematical model that could be associated with the descending dialectic method (see[PAR 86]).


In effect, in philosophy and many fundamental problems that lead to repeatedmeditations, we know today that an answer, as Heidegger wrote once, is often onlythe final step in questioning and that this process is not terminated by this response.This situation has, in fact, been known since the time of Kant. It generalizes theobservations of this philosopher around three ideas central to reason and whichcontain within them classical metaphysics – the self, the world and God – ideas thatare absolutely transcendental, according to Kant, and not simply concepts ofunderstanding. As they cannot be associated with any tangible intuition, they alreadyexceed, in effect, any form of experience. What may not have been sufficientlyemphasized with respect to this point of view is that Kant was a contemporary ofLambert7 and also of Legendre.

The question of transcendance, in mathematics, would rapidly become moreprecise. At the time of Legendre, no one knew of the existence of transcendentalnumbers. But about 40 years later, the mathematician Joseph Liouville (1809–1882)discovered that all irrational numbers that possess very good rational approximationsare transcendentals, for example:

10&1! + 10&2! + 10&3! + ... = 0.1100010000...

In 1874, Georg Cantor (1845–1918) gave his fantastic demonstration (based on theuse of the diagonal procedure) of the existence of an uncountable set of transcendentalnumbers, while also affirming, correlatively, the uncountability of the set of algebraicnumbers.

1873 then saw an important breakthrough in Number Theory, when the Frenchmathematician Charles Hermitte (1822–1901) developed methods that would allowhim to prove the transcendental nature of e, the number that comes in as the base ofNapierian logarithms.

The result for " was obtained by the German mathematician Carl Louis Ferdinandvon Lindemann (1852–1939), who was Hilbert and Hurwitz’s professor at Knisberg,before he left to teach at Munich. In 1882, in a short article devoted to the number "8,he demonstrated his famous theorem.

THEOREM.– [Lindemann, 1882] " is transcendental. As J.-C. Carrega thendemonstrated, this result, attached to Wantzel’s characterization, “makes it possibleto definitively conclude the impossibility of squaring a circle” [CAR 81, p. 3].

7 See, however, [DEB 77].8 “Uber die zahl "” (On the number "), Mathematische Annalen, 20 (1882), pp. 213–225.


Lindeman’s result is obtained based on the following principle:

Proof (abridged). Let c1, c2, ..., cn, be the complex algebraic numbers, which arelinearly independent. Then there is no equation of the type:

a1ec1 + a2e

c2 + ...+ anecn = 0

in which ai are the algebraic numbers that are not all null. If, in this, equation, wetake, for example, n = 2, c1 = c, c2 = 0, we obtain:

a1ec + a2 = 0

Hence:

ec = #a2a1

which is impossible and, thus, proves the transcendence of e. The only possible wayof satisfying this equation (positing c = i") would then make it obligatory, accordingto Euler’s famous equation (which will be discussed later on) that ei! = #1; we mustalso posit that a1 = a2, which is contrary to the hypothesis. We thus also obtain thetranscendance of ". !

We have, since, also been able to demonstrate (Gelfond, 1929) that e! = i&2i! istranscendental. On the other hand, the status of "e is still unknown.

As e is transcendental, the numbers e" and e+" cannot both be algebraic. But westill do not know whether e" or e+ " can or cannot be rational.

In a general manner, our knowledge of transcendental numbers is still extremelylimited, which could, perhaps, explain why philosophers have generally ignored theproblem and have not given much thought, since Kant, to transposing these methodsto philosophy.

Today, we can risk simply saying that the modern theological idea according towhich the existence of God (this “transcendental” being) cannot be proven in any wayis a sort of philosophical equivalent of Lindemann’s result: being inaccessible to logic,which we know admits algebraic interpretation, God – the “infinite circle”, if we wereto believe Nicholas of Cusa – can also never be the solution to any algebraic equation.

PART 2

Mathematics BecomesMore Powerful



Greek decadence coincided with a more or less obscure period, where mathematicswas concerned, at least in Europe. At this time, the Greek and Hellenistic legacy wastaken up mainly by Arab mathematicians, who also benefited from contributions madeby Indian, Chinese and Babylonian mathematicians. Because of a wide-ranging andimpressive translation effort, Euclid, Ptolemy, Apollonius, Archimedes, Diophantus,Diocles and Pappus became accessible to the Arab world, which then developed itsown research. Their studies were particularly rich and flourished most particularlyaround the 10th and 11th Century A.D., even though it is not possible to measure theimpact of this research on Arab philosophy of that time.

In the 12th Century, two mathematical treatises published by the Persianmathematician Al-Kwarismi would have a decisive influence on Europeanmathematics. One of these, according to the only extant Latin translation, passed ondecimal numbering. The other, Kital f’il-Jabr wa’l-muqabala (the Book onRestoration and Confrontation), which deals with the manipulation of equations, isthe origin of algebra, even though the Greeks were able to anticipate certain aspectsof this work [VIT 05].

Techniques for resolving equations of the third degree were implemented andeven though the Arabs failed in their research for general methods of resolution usingradicals, these were immediately applied to ancient problems (doubling a cube,trisecting an angle, constructing a regular heptagon). Other avenues were alsoexplored, such as the approximate resolution of two conic sections throughintersection as well as an early classification of equations based on the sign of theircoefficients.

About 150 years after Al-Kwarismi (the early 9th Century), computationtechniques for decimal systems were extended to polynomials and it is even likelythat al-Samaw’al, Al-Kwarismi’s successor, demonstrated the binomial formula. Hementions, in passing, that this formula could also be extended indefinitely, thanks to


the rule governing the composition of coefficients that are called “Pascal’s triangle”in the present time. This, therefore, is probably one of the first inductions of the finitetype [RAS 97].

The rules governing the calculations carried out on monomials, the rules on thedivisibility of polynomials, approximation techniques for polynomial quotients orsquare roots using negative exponents – all these also date to this period. For the firsttime, we see the appearance of a synthetic presentation of polynomials in the form ofa coefficient table of monomials arranged in decreasing order, as well as the rules tocalculate fractional exponents.

Algebra was also used for rational indeterminate analysis (rational Diophantineanalysis) in order to find rational solutions to problems that contain more unknownsthan equations. Abu Kamil, in particular, explored second-degree problems andlinear systems, using procedures such as changing the variable and eliminating bysubstitution even in this early period.

Extending the work done by the Greeks and the Indians, Arab mathematicians,who were particularly creative, also developed techniques for numerical analysis tocalculate numerical equations or the extraction of square roots. Using trigonometry,they were already in possession of interpolation methods.

In conjunction with cryptography problems, there also began a new line ofcombinatorial reflection, especially enumeration formulas (number of permutationsof n elements; number of words of n letters, one of which is repeated k times;number of words of n letters where the ith letter is repeated k times, number ofcombinations of p elements among n, etc.). The combinatorial dimension thusbecame an essential part of any mathematical work, as with al-Kashi, or the subjectof entire treatises, as with Ibrahim al Halabi [RAS 07, pp. 147–164].

Inspired by Euclid, Diophantus and Nicomachus of Gerasa, a veritable numbertheory was also developed, including reflections on perfect numbers, amicablenumbers, and the Chinese remainder theorem. This made it possible, avant la lettre,to write the large results that characterize prime numbers such as the Wilsontheorem1 [RAS 97, p. 91] or Pythagorean triplets.

Extending the work of the Greeks, again, Arab mathematicians perfected thecalculation of areas and volumes, as well as isoperimetric problems (Heron’s

1 In today’s language, Wilson’s theorem affirms that an integer, p, that is larger than 1, is primeif and only if the factorial of p" 1 is congruous with –1 modulo p. This result, however, is stillrather anecdotal and cannot be tested for a prime.


formulas) and refined calculations using numerical analysis, creating new formulasfor cones and truncated pyramids. Taking Archimedes’ work further, they createdinfinitesimal techniques. Notable in this field were the brothers Banu Musa, authorsof a treatise on the measurement of plane and spherical forms. A successor, Thabitibn Qurra, is likely to have divided the area under a parabola into trapeziums,perfectly analogous to what we would one day call “Riemann sums”. We can alsofind, in Ibn al-Haytham’s work, all the elements for computing integrals usingDarboux sums (framing, squaring the divisions with the errors reduced as much aswe wish). Arab mathematicians, however, stopped using computing areas andvolumes that could be expressed as functions of areas and volumes that were alreadyknown [RAS 97, pp. 106–112].

The computation of the areas and portions of circles, the notorious lunes and thequestion of isoperimeters (which figure would have the smallest area, keeping theperimeter constant?) did not escape them. Neither did the construction of curves,transformations and projections (affines and even projectives), or the question offundamentals (reflection on Euclid’s fifth postulate, trigonometry or geometricaloptics).

Thus, in brief, according to the specialists of the Arab world, mathematics madegreat progress on the other side of the Mediterranean, and this during those centuriesthat Europe slumbered.

The Latin West, however, seems to only have had a partial knowledge of thesedevelopments. Western thinkers were able to access certain texts because of the directcontact with Andalusian civilization through the translation of texts into Latinthrough Hebraic tradition and the exodus of Byzantine thinkers beforeConstantinople was taken over. Al-Kwarismi’s decimal writing and Indiancalculations soon became known and al-jabr soon gave a name to an entirediscipline: algebra. From this point of view, it was undoubtedly Liber abaci, byLeonard de Pise (or Fibonacci), that introduced the Latin West to Diophantus’ work,as well as the many concepts borrowed from Arab sources. However, the transferremained partial as many texts were not known, others did not arouse enough interestin Western scientists and yet others were too difficult to be translated yet. From allthe wealth of Arab mathematics, what the Western world would, in reality, assimilatewas only the basic steps, especially in the field of algebra.

From the 16th Century onwards, the West would, moreover, create its own traditionwith the German school (Christoff Rudolf), the Italian school (Luca Pacioli, Tartaglia,Cardan, Bombelli) and, finally, the French Symbolists (Viète and Descartes). Thanksto the Arabs, nonetheless, the Greek texts of Euclid and Archimedes, enriched bytheir own contributions, would be the starting point for renewed work in geometry andwould influence mathematicians like Witelo and Regiomontanus [ALL 97, p. 219].


Despite this, however, many texts would remain unknown and no Europeanphilosopher of this age seems to have been influenced, in their speculative thought,by these extraordinary Arab discoveries.

In addition, Arab mathematics itself declined from the 12th Century onwards andits contributions from this period onward became negligible [MER 11]. Westernphilosophy, especially that of the Classical Age, would thus essentially be linked tothe sui generis discoveries by the authors of this period, mathematicians andphysicists as well as philosophers.

In this, Part 2 of the book, we have arrived at the dawn of the 17th Century inFrance. We will, thus, only try to respond to three concomitant questions:

1) how did geometry, once it became analytical geometry2 revolutionize the Greeklegacy and what were the philosophical consequences of this? We will see thatDescartes’ philosophy was, in a large part, a result of this mathematical advance,which was related to the introduction of a coordinate system (what would one day becalled “Cartersian coordinates”) and the general method that accompanies it;

2) we will also explore how analysis became infinitesimal analysis, thusconfronting the problem of infinitely small quantities and their status. This would,later, be eminently discussed by mathematicians up until the 19th Century in Carnotand Cauchy’s work. We will see that Leibniz, who, along with Newton, playedan essential role in the invention of infinitely small quantities, would introduce aphilosophy that was directly related to this and which could not, moreover, have beenconceived of without them;

3) we will see that the conjunction of the existence of transcendentals, discoveriesrelated to logarithmic and exponential functions and the invention of complex numbersall led, in Euler’s formula, to the first image of unity of the mathematical sciences,while also having diverse effects on philosophy. With complexes in particular, algebraprogressed in a synthetic way that even a philosopher like Hegel (who was hostile tomathematics, which to him was essentially analytics) was forced to recognize. Later,as these numbers were more and more widely used in physics to represent phenomenain continuous media, certain physicists would have a new image of the world, a morefluid one than before. Bergson, for better or for worse, would draw metaphysicalconsequences from this.

2 We will not, here, study the history of analysis itself, that being a problem for historians. Thisquestion has, in addition, already given rise to several research projects. For example, refer to[HAI 00] and, for the Newtonian period, [PAN 05].

4

Exploring Mathesis in the 17th Century

Descartes seems to have had a vision, early on, of a large-scale reform that wouldaffect all aspects of knowledge – an idea of a “general science” that would explain“everything that may be found on order and measurement, approached independentlyof all application to any specific” (Regulae, 4).

This vision, however, remained a little mysterious. First of all, it is not certain thatit ever quite matched the confidence seen in the Discourse on Method (second part),where he declares that his aim was not to learn “all of those particular sciences thatare commonly called mathematics” but that seeing “the different ratios orproportions” that were found here, he thought it was better to limit himself toexamining these “in general”. Further, the idea of “explaining” can, clearly, takeseveral paths, not all of which are formal. Thus, it is not even certain that the ideacontained the hypothetico-deductive form that we like to think we know, that isusually associated with the rationalism of the Classical Age or, a fortiori, itsreappearance in the form of logic external (Husserl) or internal (Peano, Frege,Russell, Carnap) to mathematics, as a discussable construct. This revived the projectas a philosophical undertaking. But, above all, through this categorization of the realbased on imaginative logic that it seemed to include, it appeared to suggest an imageof mathematics that was very different from that of the set theories and,consequently, seemed much closer to what would one day be the theory of categories[RAB 09]. In Metaphysical Meditations, Descartes himself promoted the analyticalorder, relegating a synthetic presentation of his propositions to Secondes Réponses(Second Responses).

Moreover, as Vincent Jullien writes, “there are many subtle and eruditeinterpretations of what we may understand by Mathesis Universalis and we know thepossible candidates: this could be the algebrization of geometry, or the method or thetheory of proportions, or natural wisdom, or a return to the ancient tradition alreadyexamined by Proclus, up to the discussion of the debates in the 16th century relatingto certitudo mathematicarum” [JUL 09].



As the author recognizes, it remains for these responses to come together arounda common theme: order and measurement, which remain the “most stable” attributesof Mathesis Universalis.

In addition, whether we wish it or not, order and measurement arise out ofmathematics. It would be hard, to be honest, to understand where the idea of such ascience could emerge from unless it had as inspiration and guide Descartes’mathematics, that is, the science called Geometry. Whether we consider thepublications that discuss and elaborate on this science [MAR 07] or, more preciselytheir content, there is no doubt at all that it is at least a common thread.

4.1. The innovations of Cartesian mathematics

Let us recall first of all the chief contribution of Cartesian Geometry. Up untilthe 17th Century, it was commonly held that geometry was concerned with spatialfigures (lines, surfaces and solids) and that arithmetic (or algebra) was concerned withnumbers (known and unknown).

Although Descartes employed neither the term analyze1 nor analytical geometry2,

1 This word has a long history in mathematics and has changed meaning several times. A termfrom medieval Latin and derived from the Greek analuô (to untie, undo), it initially signifiednothing other than a series of successive reductions of a theorem or a problem to a theorem, ora problem that was already known (see [HEA 21, p. 291]). Introduced by Theon of Alexandria,we find it again in 1591, penned by François Viète in the expression analytic art (see the title ofhis work: In artem analyticem Isagoge), which means algebra. Viète preferred this term to theword “algebra”, which had no meaning in any European language. Once the theory of functionsand differential calculus were developed, Lagrange (1791) and then Cauchy (during his courseat the Ecole Polytechnique) would use it in the title of their works in its current sense.2 Michel Rolle [ROL 09] seems to have originated this expression. See [BOY 56, p. 155]; inthe same book, Boyer also refers to Netwon’s treatise, titled “Artis Analyticae Specimina velGeometria Analytica” [NEW 1779, pp. 389–518]. However, the expression is not attribuableto Newton himself, but to a copyist, William Jones. The opuscule edited by Horsely was,in reality, none other than ‘De Methodus serierum et fluxionum’, written in the winter of1670-71 (voir [WHI 81]). In 1797, Sylvestre François Lacroix (1765-1843) wrote in his work,Traité du calcul différentiel et du calcul intégral (Treatise on differential calculus and integralcalculus), that there was a way of conceiving of geometry that could be called ‘analyticgeometry’. This consisted of deducing properties of extrapolating a small number of principlesusing truly analytical methods. A contrario, in ‘The work of Nicholas Bourbaki’ [DIE 70,p. 140], J. Dieudonné argues against this appellation, saying that the use of ‘analytic geometry’to designate what is actually linear algebra with coordinates (a usage that spread to many basictextbooks, unfortunately) was wrong. He wished to reserve this expression for the theory ofanalytical spaces (one of the most profound and difficult theories in all mathematics).

Exploring Mathesis in the 17th Century 77

nor even coordinates3 and it must be recognized that he was the first to systematicallyassociate the following three factors:

1) the expression of a geometric reality by a relation between variable quantities;

2) the use of coordinates (even though this word is not used);

3) the principle of graphical representation.

We can see that each of these three factors interacted quite soon in the developmentof Geometry. However, before Descartes, they were almost never brought together.

Of course, from oldest antiquity, astronomical observations had led to localizingdirections in space through angular coordinates: the height above the horizon and thedistance with respect to the meridian. And the relations between these coordinateswere also soon highlighted. Nonetheless, these were practices that had almost noconnection with the science of geometry.

On the contrary, the Greeks would come in and strike at the heart of geometry,with a calculation carried out on two variables in order to characterize geometricrealities and establish geometric properties. We saw this emerge with Archytas, thenMenaechmus, in the problem of doubling a cube. Later, Archimedes, and especiallyApollonius, developed such a calculation systematically for the study of conics,whose equations Apollonius explicitly wrote in terms of oblique coordinates.

Finally, we can mention again, as the precursors to the principle of graphicalrepresentations, the work carried out by Nicholas Oresme in the 14th Century. Ineffect, to study certain phenomena, especially the movement of mobiles, Oresmetraced graphs in which he clearly distinguished a latitudo and a longitudo, whichcorrespond exactly to the abscissa and ordinate in a representation using angularcoordinates. This manner of working is the inverse of that of the Greeks, as Oresmedoes not begin with a geometric reality, but expresses a relation between quantities ingeometric form. The conception of such a correspondence must, therefore, beconsidered to fall within the framework of ideas that are at the heart of “analyticgeometry”. Nonetheless, Oresme’s views, despite being widely accepted, were in noway connected with the “analytic” practices of the Greeks. The West had come toknow of these toward the end of the 16th Century with the publication in Latin ofArchimedes and Apollonius’ work. Descartes, at any rate, seems to have discoveredthese ideas independently.

3 The word, this time, is from Gottfried Wilhelm Leibniz who also used the term “coordinateaxes”. He explicitly used this expression in “De linea ex lineis numero infinitis ordinatim ductisinter se concurrentibus formata, easque omnes tangente, ac de novo in ea re Analysis infinitorumusu” [LEI 92].


Descartes’ discovery was, in fact, a result of a desire to optimize the resources ofthe human mind. In particular, he wanted to save the imagination needless fatigue andthus he came up with the means to express geometric relations (between lines andcurves) through the intermediary of algebraic equations.

This project assumes, first of all, a revolution in symbolism, the introduction ofnew symbolic writing that contributed, as it often does, “to invention in mathematicsitself” [SER 09, p. 1205]. Descartes was, in any case, the first to use Latin letters fromthe end of the alphabet (x, y, z) for unknowns and letters from the beginning of thealphabet (a, b, c) for known variables. He also introduced or, at any rate, systematizedthe notation for numerical exponential powers using formulas of the type xn, eventhough he did still continue using xx instead of x2. Even though the notations heused in Geometry, in 1637, are not homogenous, it is undeniable that he replaced theold “diophantus-cossis” system of representation [SER 98] with a new system thatwould then be adopted by the mathematical community and largely remains in useeven today.

Within mathematics proper, the application of algebra to the geometry of curves -his grand discovery – allowed him to use purely algebraic calculations to work on thegeometry of the ancient world. Descartes’ Géométrie would thus only work withmeasurable quantities. Hence, the first sentence in the text, which is essential: “Allthe problems of geometry can easily be reduced to such terns that afterwards oneonly needs to know the length of a few straight lines to construct them”. AsGilles-Gaston Granger observes, Cartesian geometry is thus reduced, from the outset,to a metric geometry; the science of order and measurement is thus essentiallyresolved into a science of measurement since order, for Descartes, never figured as asubject of mathematical reflection. Contrary to Leibniz’, who dedicated reflection toorder (with regard to the combinatorial or Analysis situs), he had only a singlemetamathematical and purely methodological observation: order is simply that whichmethod must observe [GRA 88, p. 49].

In this context, Descartes’ mathematics was first dedicated to the resolution ofproblems using lines and circles, to which he then applied algebraic procedures. Theuse of coordinates then allowed him to unify the study of curves, putting an end to adistinction that dated back to Antiquity. Finally, in the Geometry or through hiscorrespondance (notably with Fermat and Roberval), he approached differentproblems or difficulties (constructions and resolution of certain equations of a higherdegree, methods to determine the normal to certain curves: conchoid, folium, cycloidovals, etc.) and would lay out certain theorems, such as the theorem that established arelation between the radii of four circles tangent to each other, demonstrated in 1643.


4.2. The “plan” for Descartes’ Geometry

Let us now explore Geometry in greater detail. The work that bears this title iscomposed of three large volumes:

1) Book I is dedicated to problems whose solutions can be constructed usinga scale and compass, that is, problems in 2D plane geometry, which arise fromthe theory of proportions. Among these problems, Descartes highlights the Pappusproblem, which was suggested to him by Golius, a geometrician in Leiden. Thisproblem would be the starting point for the general method that he invented;

2) Book II, titled “On the Nature of Curved Lines”, was one of the firstconsequences of the method introduced earlier, which led to Descartes proposing ageneral classification of curves that was very different from that of the ancients andled to him pronouncing which curves were admissible in geometry, which was a veryimportant and particularly problematic point, as will be seen. Indeed, as concerns thistopic, we will see that the mathematician excluded, for reasons that must be explained,certain categories of curves that were admitted by Leibniz, for example. As will beseen later on, this would have considerable metaphysical consequences;

3) Book III undertakes the construction of solutions for solid problems orsupersolid problems, that is problems whose solution brings into play conic sectionsand geometric sites that are even more composed or linear sites.

In this section, we will restrict ourself to studying the central part of Book II,concerning the classification of curves that are admissible in geometry.

4.3. Studying the classification of curves

Book II begins with the following text:

“Which are the lines that may be admissible in Geometry? The ancients werefamiliar with the fact that the problems of geometry may be divided into threeclasses, namely, plane, solid, and linear problems. This is equivalent to saying thatsome problems require only circles and straight lines for their construction, whileothers require a conic section and still others require more complex curves. I amsurprised, however, that they did not go further, and distinguish between differentlevels of these more complex curves, nor do I see why they called the lattermechanical, rather than geometrical.

If we say that they are called mechanical because some sort of instrument has tobe used to describe them, then we must, to be consistent, reject circles and straightlines, since these cannot be described on paper without the use of compasses and aruler, which may also be termed instruments. It is not because the other instruments,being more complicated than the ruler and compasses, are therefore less accurate, for


if this were so they would have be excluded from mechanics, in which the accuracyof construction is even more important than in geometry. In the latter, exactness ofreasoning alone is sought, and this can surely be as thorough with reference to suchlines as to simpler ones. I cannot believe, either, that it was, because they did notwish to make more than two postulates, namely, (1), a straight line can be drawnbetween any two points and (2) about a given center a circle can be described passingthrough a given point. In their treatment of the conic sections, they did not hesitateto introduce the assumption that any given cone can be cut by a given plane. Nowto treat all the curves which I mean to introduce here, one additional assumption isnecessary, namely, two or more lines can be moved, one upon the other, determinedby their intersection of other curves. This seems to me in no way more difficult.

It is true that the conic sections were never freely received into ancient geometry,and I do not care to undertake to change names confirmed by usage; nevertheless,it seems to very clear to me that if we make the usual assumption that geometry isprecise and exact, while mechanics is not; and if we think of geometry as the sciencewhich furnishes a general knowledge of the measurement of all bodies, then we haveno more right to exclude the more complex curves than the simpler ones, providedthey can be conceived of as described by a continuous motion or by several successivemotions, each motion being completely determined by those which precede; for inthis way an exact knowledge of the magnitude of each is always obtainable.

Probably the real explanation of the refusal of ancient geometers to accept curvesmore complex than the conic sections lies in the fact that the first curves to whichtheir attention was attracted happened to be the spiral, the quadratrix, and othersimilar curves, which really do belong only to mechanics and are not among thecurves that I think should be included here since they must be conceived as describedby two separate movements whose relation does not admit of exact determination.Yet they afterwards examined the conchoid, the cissoid and a few others whichshould be accepted; but not knowing much about their properties they took no moreaccount of these than of the others. Again, it may have been that, knowing as they didonly a little about the conic sections, and being still ignorant of many of thepossibilities of the ruler and compasses, they dared not yet attack a matter of stillgreater difficulty.

I hope that hereafter those who are clever enough to make use of the geometricmethods herein suggested will find no great difficulty in applying them to plane orsolid problems. I therefore think it proper to suggest to such a more extended line ofinvestigation which will furnish abundant opportunities for practice”.

In this text, which is essential to understanding Descartes’ philosophy, the authorbegins with a criticism of the classification of geometric problems carried out by theAncients. He then elaborates on what must be the fundamental basis for the


admissibility of curves in geometry before ending with a redefinition of the conceptof a “mechanical curve”.

4.3.1. Possible explanations for the mistakes made by the Ancients

Descartes first begins by recalling the classification of curves according to theAncients:

“The ancients were familiar with the fact that the problems of geometry may bedivided into three classes, namely, plane, solid, and linear problems. This isequivalent to saying that some problems require only circles and straight lines fortheir construction, while others require a conic section and still others require morecomplex curves”.

This classification is exactly that which we find in the collections of Pappus(Berlin, 1876–1878). Pappus says, very precisely:

“The Ancients thought of geometric problems as being divisible into three classesthat they called plane, solid and linear. Those problems that could be resolved usingstraight lines and the circumference of circles are called plane problems, as the linesor curves that served to resolve them had their origin in a plane. But problems whosesolutions were obtained using one or more conic sections were called solid problems,as we had to use the surfaces of solid figures (conic surfaces). Then there was the thirdclass, called linear, as the construction of these solutions required other ‘lines’ thanthose we have described and whose origins are diverse and more tangled. Thus, wehave the spiral, the quadratrix, the conchoid and the cissoid, all of which have certainimportant properties”.4

Authors in the 18th Century would faithfully adhere to this vocabulary. However,Descartes challenged it for two main reasons:

– the first is that this classification masks the true nature of algebraic curves and,beyond the second degree, confounds them with transcendentals;

– the second reason is that this classification restricts the theory of functions to thethree dimensions of Euclidean geometric space and proscribes the proper analyticalstudy of curves: it denies the possibility of analytic geometry, which was Descartes’major contribution to mathematics.

4 Pappus, XXXII, vol. I., p. 55, Proposition 5, Livre III.


Moreover, this major discovery could only be made through the generalizationof the notion of “dimension”, and thus by going beyond the intuitive realism of theAncients.

“By dimension (dimensio)”, Descartes wrote in Regulae, “we mean nothing otherthan the mode and relation by which any subject is judged to be measurable, suchthat not only are length, breadth and height dimensions of the body, but weight is thedimension along which objects are weighed (sed etiam gravitas sit dimensio,secundum quab subjecta ponderantur), speed is the dimension of motion (celeritassit dimensio motus) and thus, an infinity of other things of this kind” (Règle XIV,section 16) [DES 77].

In other words, the notion of dimension is detached from its concrete meanings (thethree dimensions in space) to become an abstract, conventionally designated element.At the same time, Descartes makes it implicitly possible to not only consider morethan three dimensions, but also the subsequent existence of what physicists would oneday call a phase space, that is an abstract space whose coordinates are the dynamicvariables of the system being studied.

In a correlational manner, the presence of an exponent 2 or 3 in an algebraicequation would no longer be the same as surfaces or cubes. As Descartes says at thebeginning of the Geometry (I, I), “it must be noted that by a2 or b2 or similarnotations, I only ordinarily mean very simple lines, although, to make use of thenames used in Algebra, I name then squares or cubes etc.”.

How is this possible? Precisely because the introduction of that which we nowcall “Cartesian coordinates” or, more generally, the idea of a “Cartesian coordinate”(which does not explicitly figure in the text, although the procedure is present) makesit possible to define any kind of correspondence between the variables x and y,regardless of the exponent of x or of y. In other words, in all cases the graphs of thesecorrespondences take the form of lines, which takes away from the concretesignificance of their exponents (surface, solid, etc.).

From here, all other justifications for the classification proposed by the Ancientsbecome invalid. Nothing, at any rate from instruments or mode of construction, canpermit their justification. Therefore, a new classification of curves, one that is morerational, must replace this older one.

How far is this classification rational? Why was it not truly satisfactory, leading, aswe will see, to an impasse for Descartes the geometrician and, consequently, Descartesthe philosopher? This is what we shall examine in the following section.


4.3.2. Conditions for the admissibility of curves in geometry

What are the conditions that curves must satisfy in order to be called “geometric”?Descartes listed four conditions:

1) traceability: geometric lines must originate from other geometric lines. To tracethem, we must thus only assume the possibility of two or more lines moving or pushingeach other;

2) the series of intersections of these lines (which are series of points) mustcorrespond to other lines;

3) exactitude: here, that which is precise and exact is, in general, said to begeometric, that which is not is said to be mechanical. Consequently, we mustnot exclude composite lines from geometry just because they are composite lines.However complicated, they may still be perfectly exact;

4) the continuity of generative motion: in truth, all lines must be included ingeometry, whether complex or simple, if they can be described by a continuousmovement or by a succession of movements that follow each other.

What is the significant of these conditions from a mathematical point of view?In reality, they signify that Descartes restricted the notion of function (or functionalcorrespondence) to that of exact proportion.

What consequences would this restriction bring about?

Even as Descartes, refuting the completely empirical classification of theAncients, brought about a considerable extension of the concept of the algebraiccurve, by generalizing the concept of dimension and associating polynomials of anydegree with curves, he restricted the limits of geometry by excluding certain curves(that he called “mechanical”) from it. His usage of “mechanical” was not the same asthat of the Ancients, but based on a particular meaning that he gave to the word,namely, curves whose determination is inexact or those that cannot be defined by acontinuous movement. These curves are, in fact, what we today call “transcendental”.

Let us recall that those curves that we now call “transcendental” are those whoseequation is not algebraic, that is, whose equation is not a polynomial (knowing thatany polynomial equation admits algebraic numbers as solution, but that there existnumbers such as ", e, etc., which are not the solution to any algebraic equation). Asthese numbers are called “transcendentals”, the curves where they figure haveinherited this appellation.

According to what has been said, Descartes would then admit curves of anydegree, as long as they were algebraic, but would reject from geometry any reflectionon spirals, quadratrices and, generally speaking, all transcendental curves. Before


attempting to understand this strange ostracism, let us quickly review this type ofcurve.

An index of the large families of curves (planes) would today show uptranscendentals as structured in the following manner:

– quadratrix (Hippias, 460 B.C.), a curve that makes it possible to trisect an angle;

– Archimedes’ spiral

– logarithm;

– logarithmic spiral;

– trisectrix and MacLaurin’s trisectrix;

– semicubic parabola (Neil, 1659, Leibniz, 1687);

– tractrix (Huygens, 1692);

– sinusoids (developed with trigonometry by Indians, Greeks and Arabs). Thesinusoid is the planar projection of a helix;

– trochoids (a very general family of curves defined by the location of a fixed pointon the circumference of a circle that rolls along a straight line);

– epitrochoids and hypotrochoids (comprising the remarkable subclasses of theepicycloids and hypocycloids). The cycloid was discovered by Galileo (1599) andMersenne5 and cycloidal curves were studied by the astronomer Roemer in 1674. Thegenerating of these curves was studied by Daniel Bernoulli in 1725 and astronomersoften found them in star coronas. They are also visible in optics (in caustics);

– epicycloids themselves admit, as special cases, curves such as the cardioid andnephroid (Huygens, Tschirnhaus, 1679);

– hypocycloids admit, as special cases, curves such as the deltoid (Euler, 1745) or,again, the astroid.

5 Marin Mersenne (1588–1648), a Minim Friar destroyer of deists, atheists and other libertines.Known essentially for his correspondence with Descartes and, beyond Descartes, any Europeanthinker of the time. He carried out some mathematical work, traces of which remain (Mersenne’sprime numbers, of the type 2p"1), and also some work in physics (studying the field of gravityusing a pendulum, telescope with parabolic mirror, acoustics of the propagation of sound, etc.).His best-known work is a treatise in musicology, L’harmonie universelle (Universal Harmony)[MER 36], a veritable sum of the musical knowledge of the time, but without any real link tocosmology, unlike Pythagorism, from which he parts ways.


Descartes, thus, removes transcendentals from the sphere of considerations.Why? We have answered this by mentioning the inexactitude of the relations. Moreprecisely, we can say that transcendentals fall outside of what Descartes consideredto be legitimate constructions in geometry.

4.4. Legitimate constructions

The general form of the legitimate construction is explained by Descartes at twoplaces in his Geometry, namely at the beginning of Books II and III. The procedure forgenerating curves, that is described in these texts, assumes a series of mobile squaresof the following form.:

Figure 4.1. The construction of curves

To construct curves of a higher kind, Descartes proceeded as follows: consider thelines AB,AD,AF and so forth, which we may suppose to be described by means ofthe instrument Y Z . This instrument consists of several rulers hinged together in sucha way that Y Z being placed along the line AN the angle XY Z can be increased ordecreased in size, and when its sides are together, the points B,C,D,E, F,G,H , allcoincide with A; but as the size of the angle is increased, the ruler BC, fastened atright angles to XY at the point B, pushes toward Z the ruler CD which slides alongY Z always at right angles. In a like manner, CD pushes DE that slides along Y Xalways parallel to BC; DE pushes EF ; EF pushes FG; FG pushes GH , and so on.


Thus, we may imagine an infinity of rulers, each pushing another, half of them makingequal angles with Y X and the rest with Y Z6.

J. Vuillemin [VUI 60, p. 84] offers the following commentary:

“For each position of the ruler Y X , that is, for each opening of the angle XY Z, apoint on the curves AB,AD,AF,AH... is determined. The choice of theseparticular points is entirely arbitrary and made at will or, as Descartes, says, isindifferent. Consequently, no points are prioritized in the construction. This alsomeans that when we intrapolate between two points determined in this way, or whenwe extrapolate from a defined segment of the curve, our procedure is entirelydetermined and governed by the relations that can be expressed in a finite number ofalgebraic operations with the knowledge of other points. No indetermination remainshere and our construction is perfectly exact as it fulfills the double condition of beingentirely regulated and continuous”.

The essential point that must be noted here is as follows: Descartes reduces themeaning of functional relations to relations of proportionality. For Descartes, asJ. Vuillemin writes [VUI 60, p. 88]:

“A relation is functional if it makes it possible to bring a given length intocorrespondence with another length deduced from the first through a finite number ofalgebraic operations. Only such a relation, according to Descartes, can be used in aconstruction so as to attain all the points on a curve without excluding any. Thispossibility will ensure the chaining and intuitive continuity of the curve, without theneed to bring in infinite considerations. The line y = ax+ b or the parabola y = ax2

are thus representable as exact proportions that I can comprehend intuitively orthrough deduction”.

6 This assembly of rules and set squares, sliding one over another, makes it possible to describeincreasingly complex curves. Using modern notations, the point B describes a circle of radiusR and x2 + y2 = R2, the point D describes the curve of equation y = x2

R , F describes thecurve of equation y2 = x3

R , the point H the curve of equation y3 = x4

R , etc. All these curvesare geometric (as opposed to the mechanical “transcendentals”). As this method still seemedinsufficient to Descartes, he then unveiled his master idea: distinguishing curved lines assumesthat we know the relation of their points to those of straight lines (what would one day be calledthe “Cartesian coordinate”). That is, we must know the equation of the curve with respect toa system. The classification of the curves would then be based on the degree of the equation.Refer to Descartes, Geometry, p. 318 onwards for an illustration of this idea (“the manner ofdistinguishing all these curved lines into certain genres and of knowing the relation that all theirpoints have to those of straight lines”).


Conversely, the points of logarithmic and transcendental curves cannot all beattained. Here again, J. Vuillemin can clearly see the problem [VUI 60, pp. 84–85]:

“Let us consider, on the contrary, in the Cartesian construction of the logarithmic,the determination of two ordinates JK and PQ; they are commensurable, with theordinate of origin, AH, which is given. Consequently, up until now, the mode ofdetermining the curve, point by point, is the same as earlier. but if we move on to theordinate V U , an infinite process is required; the construction ceases to be exact andbecomes approximate as it is neither governed by a finite proportion with points thatare already known nor, consequently, is it regulated in a continuous and finite relationwith these. Intrapolation and extrapolation thus become fumbling according toDescares, as they envelop the infinite.

In the case of the transcendentals, if we make an exception for the very particularpoints that can be constructed in an entirely different manner (as, in logarithmic, by theconstruction of commensurable ordinates) as they respond singularly to proportionsthat are more or less complex but always exact; the other points, who number is, inmodern language, an uncountable infinity, can only be ’determined’ in appearance bythe movement, because the movement is either discontinuous or is joined to a secondmovement, but without being linked to it by a rule, that is an exact proportion”.

4.5. Scientific consequences of Cartesian definitions

It is true that Cartesian mathematics is not reduced to geometry – far from this. Inhis correspondence, notably, Descartes demonstrates a strong and detailed knowledgeof curves that did not find mention in his Geometry because of the retractions that wehave pointed out.

The consequences of reducing functionality to proportionality and the exclusionof transcendentals are no less considerable for Cartesian science. The only curves thatcould have had a possible application in the field of life sciences and natural sciencesare thus excluded. The logarithmic, which would become so important later on inpsycho-physics, with the Weber–Fechner law, as well as the logarithmic spiral, whichhas many links with the golden number and phyllotaxy, was abandoned. Thus, themathematics of sensations, like that of morphogenesis, would need to wait a few morecenturies to emerge and gain legitimacy. Cartesian science, qualitative and ill-founded,would chiefly remain imaginary.

Another paradoxical consequence: Descartes, who was such an excellentmathematician otherwise, could not invent the infinitesimal calculus, even though wefind, in his work, an anticipation of this invention and several substitutes for it. Hereagain, the fear of encountering problems that threatened to go beyond the ability tounderstand introduced limitations that were, in a sense, almost paralyzing.


4.6. Metaphysical consequences of Cartesian mathematics

Even more important are the metaphysical consequences of this situation:

1) restricting geometry to only considering what is “’precise and exact”corresponds, in metaphysics (that is, where Mathesis, free of the idea ofmeasurement, remains uniquely governed by order) to the restriction ofunderstanding to the consideration of only clear and distinct ideas. There is, ofcourse, no question of denying the existence of other realities. But it is useful to try,as far as possible, to get from all things, including those that will forever remainobscure and confused, a clear idea which separates them from the “clear”7. Descarteshere remains negative as, unlike Leibniz, he was unable to create a real theory ofconfusion or clarity–confusion, as such;

2) thinking clearly and distinctly, that is, thinking in metaphysics in as exact andprecise a manner as we think in mathematics, is thus the same as clearlydistinguishing between realities that we strive to comprehend by thought and,thereupon, try to break down into as many simple elements as possible, that can becomprehended by intuition. It is only after this that we can envisage the complex as acombination of simple elements. This is how Descartes’ method is explained inphilosophy as well as his notorious dualism that seeks to first separate as far aspossible (at least from a methodological point of view) these highly dissimilarrealities of the soul and body, even though we can only know this through theintermediary of the body–soul complex. In reality, the determination of thisintersection, as in the theory of algebraic equations, assumes first of all the study ofeach of these to “functions” and only the resolution of a system that is well formed inadvance can yield the solution;

3) the theory of proportions (see the commentary by J. Vuillemin [VUI 60, p. 119onwards]) has served as a model for Cartesian metaphysics and, more generally, forclassical metaphysics. Here, it is the concept of order that is central to debates.Moreover, just as Descartes classifies equations according to their degree inmathematics, in metaphysics, he classifies ideas based on their degree ofcomposition, starting with the simplest and moving toward the most complex. Doubtis, therefore, a regression from the most complex (which are the most fragile) to thesimplest, while the normal progression of the Metaphysical Meditations, followingthe famous “I think therefore I am”, would result in progressively reconquering thereal in all its complexity. Thus, human nature, like pure intelligence, is easier to knowthan the body. But from the simplest ideas, we can generate all others. Nonetheless,while chaining reasons is infinite in mathematics (as I can construct curves of any

7 We can find the same type of reasoning used by François Dagognet – in this sense, veryCartesian – in his study on trouble [DAG 94].


degree), in metaphysics, the number of truths would remain limited. Here again wecan find a situation in metaphysics analogous to the exclusion of transcendentals:feeling, so obscure and so complex, so approximative in its descriptions that it doesnot fit into any criteria for the clear and distinct, remains largely outside the scope ofCartesian considerations, just as transcendental curves remain outside the scope ofprecision and exactitude;

4) the fourth rule of the Cartesian methods – “to make enumerations so complete,and reviews so general that I might be assured that nothing was omitted” – is generallyinterpreted as being equivalent to the recapitulative enumeration of all the variables ofa problem. After having divided the problem (rule 2) to reduce it to simple intuitions(rule 1), as we have deductively chained these by observing the rules of classicalinference (rule 3) we must ensure that we have forgotten nothing along the way. But,if this were the case, this rule would do double duty along with all the earlier rules,especially that of division. When we reduce the size of a problem, for example whenwe reduce a fourth-degree equation to a product of two second-degree polynomials,it follows, by the very process of the method of indeterminates, that we cannot forgetany element of the problem. This remark also holds good if we give this equation aphysical meaning. In reality, for Jules Vuillemin [VUI 60, p. 137], the fourth rule isnot on the same place as the others: it concerns methods and comes up as a reflexiveprecept, which is, in a way, a regulator:

“In Geometry the fourth rule is illustrated by the examination where wedemonstrate that all solid problems can be reduced to two constructions of theinvention of two proportional means and the trisection of an angle. Descartesexamined all possible cases to which the third-degree equation could be reduced anddistinguished three among them based on the respective signs of the coefficients ofthe equation from which the second term is removed. The problem of the possibilityis thus resolved by the fourth rule inasmuch as we can exhaustively list out all theways and, from the different procedures for the solution, choose the generalprocedure”.

As Gilles-Gaston Granger would observe, this rule thus presents ametamathematical character. All the problems considered by Descartes withinmathematics itself can, thus, be reduced to algebraic combinations of length.Geometry is, indeed, reduced to a calculation, but on the basis of the presuppositionthat “there is no calculation but of quantities and no general calculation or algebra butof these abstract quantities that are pure numbers” [GRA 88, p. 49].

Thus, Descartes could not even glimpse at Leibniz’s dream: to establish acalculation that, in scope, is not reducible to measurement, which is the same asbringing the questions of order, position and situation back within mathematics itself,thereby also heralding the emergence of one of the major fields of mathematics of the20th Century: topology.

5

The Question of Infinitesimals

The construction of the concept of the infinite, initially philosophico-theological,has a long history directly linked to the evolution of science and mathematicaltechniques of computation. At first it was a word used in everyday language – moreor less fluid, as with all terms in natural language – and containing within itself purelynegative determinations; it then became, as these were eliminated, perfectly effectiveand opened access to fields of objects hitherto unknown: for example, infinitesimals,or again, the infinitely large, both of which would be formally studied in mathematics.The question that then arose was: is it possible to remove the metaphysical aspectsfrom the positive concept of infinity (traces of the metaphysical still persisted, asthe 19th Century transitioned into the 20th Century, within the science itself). Threedistinct periods can be traced in the formation of this concept:

1) the Antique period, where the principal problems of this concept were alreadybeing sketched, both in philosophical language as well as mathematical language;

2) the Classical Age (the 17th and 18th Centuries) where the concept of“infinitesimals” would be precisely defined, through the Newtonian and Leibniziandiscoveries of the infinitesimal calculus;

3) the modern period, which opens at the end of the 19th Century (in 1882, to bevery precise) and where, through the work of the German mathematician Georg Cantor(founder of the set theory), the concept of the infinitely large would be rigorouslycharacterized and definitively instated into the field of mathematics.

We will here restrict ourselves to the study of the first two periods, returning to thelast period in Chapter 9.



5.1. Antiquity – the prehistory of the infinite

As we can recall, this period dates back to the 7th or 6th Century B.C. During thisera, the infinite, a concept present in the philosophy of Anaximander (610–547 B.C.),would first appear as a simple indetermination; illimitation here borders onuncertainty, in the sense of something that cannot be experienced (see section 5.1.1).The question of irrationals, as well as that of paradoxes related to a geometriccontinuum (the famous paradoxes described by Zeno of Elea) would bring home thecomplexity of these subjects. Finally, once identified, the dual nature of infinity(actual and potential) was placed, by Aristotle, within the framework of a long debatein which mathematicians and philosophers would argue against each other, with theaim being to see if it were possible to positively characterize the infinite.

5.1.1. Infinity as Anaximander saw it

Concepts are not born fully formed. They must be considered both within thenetwork of a language and within the culture into which they are born and, over time,are associated with different types of rationality. Where we say “infinity” (otherwise,“the negation of the finite”), the Greeks use a completely different word: apeiron,which has, in fact, a double etymology. This term may be referred back to peiras, thelimit, or, with the prefix “a”, denoting exactly the absence of a limit – the limitless.But we may also compare apeiron to the verb peiraô, to experience, therebysignifying (again, with the prefix “a”) a reality that cannot be experienced: here, wemay see the idea of an uncertain perception, or that of properties that are difficult todefine1.

This loose conceptualization may be explained, in part, by Anaximander’sposition in the Greek world. Around the 7th Century B.C., general explanations ofthe universe tried to replace ancient mythologies. The major thinkers of the periodtried their outmost to link the genesis of the universe to a single “Element”. ForThales, who slightly predates Anaximander, this role was played by Water.Anaximenes, who came slightly later, thought this Element was Air. Heraclitus,much later, would choose Fire. Empedocles would add Earth to the other three andpreferred to consider all four together: Water, Air, Earth, Fire.

Anaximander’s conceptualization was, therefore, born in this cosmologicalconcept of ancient physics, and apeiron (or the infinite) seemed to him to be aprinciple with which to explain the universe, just like the Elements, strictly speaking,of the pre-Socratic thinkers.

1 These observations are borrowed from X. Renou [REN 78, p. 22].

The Question of Infinitesimals 93

This apeiron may be briefly characterized as follows:

1) as an indeterminate, and compared to Thales’ Water, it is a universal componentthat lacks all qualities. Water was a cold and wet principle that had the role ofexplaining the birth of heat and dryness, as well as all tangible qualities. On thecontrary, apeiron was characterized by the fact that it was a more general principlethan these systems of qualities and, therefore, it held even beyond their specificities,containing within itself both hot and cold, wet and dry, and all other pairs of tangiblequalities;

2) In no way is it a precursor to our infinity, as such. Apeiron signified only anoutline of physics’ quest for an invariant that would last beyond all of the processes oftransformation of this world. In fact, it is not even very closely related to “physics”.It only comes in to account for the general fate of the universe, in that it is that fromwhich everything comes and that to which everything returns. But it does not play apart in the explanation of real physical processes: lightning, evaporation and othermeteorological phenomena cannot be explained by it;

3) we thus find ourselves considering a very archaic type of thinking, whose“logic” and whose conception of an explanation is very different from ours. Forinstance, apeiron, being indeterminate, is simultaneously matter and form. Inasmuchas it is “that which makes things exist”, it is that which creates matter. But, moreover,while being that which produces their tangible qualities, that is, which makes itpossible to differentiate them, characterize them or qualify them, it is also that whichgives them form.

In conclusion, Xavier Renou suggests that this is also the case with Anaximanderexplaining his theory. He appears at a point in Greek history where the ancientdistinctions are crumbling (myth, especially, no longer has explanatory value) butwhere, at the same time, nothing has yet been found to replace them. In particular,there is no coherent conceptual system that can substitute them. There is, therefore, avoid, an indetermination of thought, that Anaximander constructs into a doctrine. Asthe author suggests, with this philosopher, an indetermination of thought itselfconstructed a school of thought of the indeterminate [REN 78, p. 24]. Thus, toconclude, infinity was first discovered in the first errant explorations of thoughtliberated from myths.

5.1.2. The problem of irrationals and Zeno’s paradoxes

After Anaximander, thinkers abandoned this world of the indeterminate which hehad made his, for a world of determination. With Pythagoras, Greek thoughtdiscovered the properties of numbers and relations between numbers. Pythagoras, aswe have seen, originated the theory of proportions (or medieties), which influencedPlato and all Greek thought. The central idea, that all things are in a ratio with eachother as numbers, that is, in proportion, suggests that the universe in general is


proportional. It is an Order (etymology of the word cosmos) and a measurable order.We thus saw the birth and development of an entire ideology of measurement, whichwould be applied at different levels: the social, moral, aesthetic, etc. Not only is theuniverse ordered and proportional, not only is this a grand order, a “macro-cosm”,but social life must also be as harmonious and well regulated as the universe and itscelestial movements. As for individuals – they must also be subject to a moral oftemperance and measurement. As we have seen, this was the “nothing in excess”principle of Greek wisdom, which also held for aesthetics: Greek art established thenorms of beauty based on the proportion of forms. Consequently, in this limitedworld, the infinite only appeared as a threatening thing that destroyed proportion,harmony and measurement. The result was excess (hubris, in Greek) – a situationthat must be prevented at all costs.

Unfortunately, as we have seen, excess, and with it, the infinite, tumbled over thethreshold the day the existence of “incommensurable” quantities was discovered –quantities that could never be brought into any proportion. Being obliged to acceptthese entities (that are today designated by the numbers

"2,"3, etc.), the Greeks

tried to transform these “proportionless” numbers (therefore called irrationals or, inGreek, aloga) into rationals. This was done either by raising them to a power (turningthem into dunamei monon rêta, as it was called) or by approximating them using aseries of rational numbers (which anticipated continuous fractions).

On the theoretical front, the concept of “number” (or at least the concept of ageometric “quantity”) would then evolve and, under the influence of mathematicianssuch as Eudoxus or Theaethetus, Plato would soon be able to present a unified concept.In Philebus, in particular, as Paul Kucharski so effectively demonstrated [KUC 51,pp. 29–59], the number appears as a mixture of the Self and the Other, of the limitedand the limitless. And if Plato could defend this conception, it was only because thePythagoreans, using a technique that anticipated continuous fractions, made it possibleto make any real number appear as that which we call, in modern terminology, the limitof two convergent series, one which approximates it by excess, the other by defect. Awhole number, or a rational number, thus became a specific case in the set of numbers.

If the Greeks had indeed encountered mathematical infinity in numbers (both withirrationals such as

"2 as well as transcendentals like ", both being non-periodic,

non-terminating decimal symbols) they did not clearly recognize this infinite and didnot characterize it well. In particular, neither were able to clearly conceive of the ideaof an actual infinity, nor were they able to conceive of it as an idea that was a passageto the limit. The best example that we can give of this is the question of the paradoxesthat overwhelmed Zeno of Elea, a disciple of Parmenides.

Let us briefly restate Zeno’s problem: initially it was only to verify theParmenidean proposition according to which the being is immobile, which heintended to prove by attempting to refute all possibility of movement. His reasoningwas quite simple, even though he dedicated several lengthy pages to expounding it.


Let us leave aside the first argument raised against movement, namely the paradoxof the arrow shot by an archer that, according to Zeno, is completely immobile at everypoint in its trajectory. Let us examine only the second argument, the famous paradoxof Achilles and the tortoise. There is a race between the Greek hero Achilles and atortoise. Achilles is very fast and thus gives the tortoise a head-start. The questionthen is: will Achilles catch up with the tortoise? Zeno says that this is impossibleand explains it thus: let us assume that the tortoise has covered a segment AB. ForAchilles to catch up with the tortoise, he would also need to arrive at B, but when hedoes this, the tortoise would have continued on its path and would be a little further,let us say at B!. Achilles would then arrive at B! but by this time the tortoise wouldbe at B!! and so on. Although the distance between the protagonists keeps shrinking,Achilles will never catch up with the tortoise, as a line is indefinitely divisible.

This is, of course, a paradox, however the question is to know why Zeno’sreasoning is specious. When we examine it we see that there are, in fact, twosophisms involved, related to a misunderstanding of the infinite:

1) The first is that Zeno reasons in terms of space and not in terms of time.Consequently, he does not give Achilles enough time to catch up with the tortoise.Let us take the following case: if, for example, Achilles runs at a speed of 2 m/secand the tortoise moves at a speed of 1 m/sec, if Achilles allows the tortoise ahead-start of one meter (during which he remains at the starting line) then he onlyneeds two seconds to be able to catch up with the tortoise. However, this is not howZeno looked at the problem. His assumption was, in substance, that Achilles leaptover 2 m, after which the distance between him and the tortoise was reduced to 1 m.He then makes another leap over 1 m in 1

2 sec. and the distance between him and thetortoise reduces to 1

2 m, etc. In this hypothesis, the time that Achilles takes to catchup with the tortoise is:

1 +1

2+

1

4+

1

8+ ...+ ... sec

This series is a geometric progression whose sum is always smaller thantwo seconds, two being the limit of this sum when it has an infinite number of terms.

Thus, it is evident that Achilles will never be able to catch up with the tortoise inless than two seconds. On the other hand, if we give him the required time, he wouldthen cover the distance of 4 m between the tortoise and himself in two seconds, that ishe would make two leaps of 2 m and easily catch up with the animal.

In other words, the paradox is, first of all, based on the fact that we do not giveAchilles enough time to cover the distance between the tortoise and himself.

2) If we go a little further, we can also see that Zeno can only support hisassertion by assuming that Achilles and the tortoise move in a discontinuous manner


up to infinity, through a series of leaps that would today be called “infinitelycountable”. But we cannot accept that Achilles and the tortoise move in this discretemanner across the field while time flows in a continuous manner. This would assumethat the continuity of the field was divisible into a series of countable parts whiletime, which is continuous, is divisible into a series of uncountable parts. However,physical time: and this would be Bergson’s major criticism – can only be representedthrough space, it is spatialized time, such that either both space and time arecontinuous or both are discontinuous. Thus, we cannot accept that one progressesindependently of the other. In fact, from a physical point of view, the two movementsof Achilles and the tortoise must be considered to be continuous, like time. This,naturally, precludes the possibility of the indefinite succession of discontinuous leapsthat Zeno presupposed [DEL 52, p. 208; BRO 26]2. We see that the paradox canactually be resolved quite easily without needing to postulate, as Bergson does inChapter IV of Matière et Mémoire (Matter and Memory), a so-called “indivisibility”of movement [BER 59, pp. 3–14].The fact remains that the opposition of thecontinuous and the uncountable would only be theorized toward the end of the19th Century.

5.1.3. Aristotle and the dual nature of the Infinite

Aristotle would clarify the situation through the introduction of a doubledistinction:

1) first, the essential distinction between two types of the infinite, potential infinityand actual infinity (and, like the Greek thinkers before him, he did not really recognizethe existence of an actual infinity);

2) second, the distinction, no less essential, between the infinite by division (whichZeno had glimpsed) and the infinite by addition or by composition (which had not beenseen until now).

The main writings by Aristotle on this question can be found in Physics III and VIand in Metaphysics M and N. However, we will confine ourselves, here, to the studyof the texts in Physics III, which are more than sufficient for our requirements.

The first question that Aristotle raises in these texts is the question of the existenceof the infinite. After having listed out (in 203 b 16) the five reasons for believing inthe infinite (infinity of time, of the divisibility of quantities, the unending character ofgeneration, the concept of limit and, lastly, the movement of thought in the numberseries, the increase in quantities or progression toward a space external to the world),

2 V. Brochard does not seem to have been very perspicacious here.


Aristotle examines, in 204 a 8, the reasons that we may have for not believing in theinfinite.

We cannot, unfortunately, study these texts in great detail here. In summary: whatwe find here is that Aristotle lays out a number of arguments against the existence ofthe infinite. According to him, the infinite cannot exist as a substance or as an attribute.Nor is it compound, nor single. Finally, it would be incompatible with the doctrine ofplace, so close to the author’s heart. Aristotle believed that everything, in effect, musthave its place and, furthermore, that there was a place for everything. This assumes,of course, that the things in question are finite, for if not, there must be an infinitenumber of places to contain them, which contradicts the idea of place.

Thus, for all of these reasons, Aristotle concluded, in 206 a 7–8, that a body that isan actual infinite (energeia) does not exist. However, Aristotle then observes that if weconversely completely deny the existence of the infinite, the consequences that wouldfollow would be unacceptable: for example, with respect to time, we would have toassume a beginning and an end to time; similarly, for quantities, we would not be ableto explain their divisibility.

We must, therefore, recognize that the infinite exists after all, but that thisexistence is an inferior existence. It is the existence of a thing “in potential”(dunamei), and not of a being “in actuality” (energeia). We can also note, in passing,that when Aristotle distinguished between actuality and potential as opposites, hegenerally used the term entelecheia. On the contrary, when he spoke of actualinfinity, the term he used was energeia. The reason for this is that actual infinity isnot exactly the entelecheia of the infinite in potential. Aristotle specified that in theexpression “infinity in potential”, the expression “in potential” must certainly not beunderstood in the sense he uses it when he speaks of the relations between matter andform. For example, when we say of a material that it is “in potential” a statue, thismeans that it may become a statue in the future. But we cannot say that a “potentialinfinity” will ever become an “actual infinity”. It will always remain potentialinfinity. As Aristotle says in Physics III, 206 a 27, “the infinite resides in the fact thatwhat we take is always new, what we take being, certainly, always, limited, butalways different”.

This results in there being a considerable difference in status between that whichwe call infinite by composition (or addition) and that which we call infinite by division:

– Aristotle believed that in reality, infinity by composition did not exist. In theprocess that is addition or composition, and where each time we add a new elementto the earlier element, we always arrive at something that is finite. And there is alimit to this operation on the cosmic level, that is, there exists a primary heaven thatenvelopes all others. Thus, as the world is a finite entity to him, Aristotle deduces theimpossibility of an infinity by composition, even potentially;


– on the other hand, the infinite by division does indeed exist and cannot existexcept in potentiality: in the sense of reduction or division, contrary to what happenswith increase or composition, we can always exceed any fixed quantity. There is,therefore, an infinity by division.

Is the absence of actual infinity and the absence of infinity by compositionharmful?

Are they, first of all, harmful to mathematics? Aristotle asked this question in207 b 27 and responded in the negative. “This reasoning does not deprive themathematicians of their study, either, in refuting the existence in actual operation ofan untraversable infinite in extent. Even as it is they do not need an infinite, for theymake no use of it; they need only that there should be a finite line of any size theywish”.

As we will see, however, there would come a time in mathematics when it wouldbe necessary to consider the infinite, and an infinite in actual operation, and when itwould be necessary to renounce Aristotle by introducing considerations oninfinitesimals in actuality as well as infinitely large quantities in actuality. In both ofthese cases it would be necessary to think of infinite series, and to consider theseobjects as axiomatically defined totalities and not as simple potential infinities.

At this point, the absence of an actual infinity in Aristotle’s theories would notprevent philosophers and theologians from engaging in a long-running debate, in laterperiods, over whether an actual infinity could exist, whether or not it contravenes thenature of the Christian God and whether or not it exists outside of God, that is, whetheror not God can create an infinite multiplicity, etc. We will not be entering into thosedebates here.

5.2. The birth of the infinitesimal calculus

The quest to determine the concept of infinity in mathematical terms would goon to the 17th Century through reflections on the infinitesimals that would come inin the development of the remarkable analysis algorithm – the infinitesimal calculus.Why did these considerations and the invention of such a calculus come about in the17th Century? There are two sets of reasons, both physical and mathematical, thatexplain this:

1) the necessity for a mathematical description of nature and the movements ofbodies in physics, which began with Galileo, led to the use of all possible resourcesof the analysis invented by Descartes, notably to find and define relations betweenquantities conceived of in a local and instantaneous manner (such as speed) and


quantities related to the idea of a space covered in a certain time, like movement,these latter concepts being defined in a general manner;

2) from the time of Archimedes, and through the question of computingquadratures all through the medieval period, people also sought to compute the areasof any given form. This required considering elements of infinitely small surfaces orvolumes such that, from the early 17th Century, infinitesimal methods, especiallyunder the influence of Cavialeri, Roberval, Pascal and Wallis, proliferated and resultsmultiplied. There then arose the necessity of assembling and ordering them.

This attempt at systematization was the work of two men, Isaac Newton(1642–1727) and Gottfried Wilhelm Leibniz (1646–1716). Both, independent of eachother, invented a handy algorithmic procedure that made it possible to work with theinfinitely small and to geometrically resolve (using the same method) problems thatwere, until then, considered isolated and unrelated to one another. Newton andLeibniz can both be rightfully considered the inventors of differential calculus andthe modern integral [DAH 82, p. 177].

5.2.1. Newton’s Writings

Newton’s writings on the infinitesimal calculus span three texts:

1) De analysi per aequationes numero terminorium infinitas (On the analysisthrough equations of an infinite number of terms) composed in 1669 and published in1671;

2) Methods of Fluxions and Infinite Series, written in 1671 but published only in1736;

3) the Quadratura curvarum, written in 1676, published in 1704.

We will here consider only the second text, which is moreover the more famousone. The other two are of lesser importance: the first, because here Newton is stillunder the influence of Barrow and Wallis and the third because it is, in fact, simply anextension of the second and we chiefly see here the emergence of a novel idea – thatof the limit, which would in any case be explicitly introduced in the Principia.

Let us, thus, explore the concept of the infinitely small, as it appeared in Methods ofFluxions and Infinite series [NEW 66]. In this book, Newton begins by distinguishingbetween two types of mathematical quantities:

1) mathematical quantities resulting from “a continuous augmentation, like thespace that describes a body in motion”. Newton calls these quantities fluents;

2) the quantities that result in the above quantities and which are, therefore, withrespect to the resulting general movement, the local velocities. Newton calls thesefluxions.


The model that inspired Newton is thus implicitly the theoretical mechanicsmodel. Moreover, the functions that Newton considers throughout the book arealways functions of time, time being a universal variable of every functionalcorrespondence in this book. He is not, however, interested in time as a concept byitself. It is the uniform flow of time that he takes into consideration and that allowshim, moreover, to introduce these concepts of fluxions and fluent quantities. Hemakes this explicit in the book in the following manner:

“Now those quantities which I consider as gradually and indefinitely increasingI shall hereafter call Fluents or Flowing Quantities, and shall represent them by thefinal letters of the alphabet v, x, y and z... and the Velocities by which every Fluentis increased by its generating motion (which I may call Fluxions [...] I shall representby the same letters pointed thus, v, x, y and z” [NEW 66, p. 21].

The fundamental problem that Newton poses, based on this, is as follows: “Giventhe relation between the fluent quantities find the relation between their fluxions. Andvice versa”.

In other words, using more contemporary terms: given a relation betweenquantities of the type “movement”, susceptible to decreasing or increasingcontinuously, find the corresponding “instantaneous velocities” and, conversely,given the “instantaneous velocities”, reconstitute the corresponding “movements”.

We thus have two problems; the direct problem (going from the curve to thetangents), in other words, that which in modern terms is called a derivation problem;and the inverse problem (going from the tangents to the curve), that is, what we todaycall the problem of finding a “primitive”.

The solution to the direct problem is made explicit by Newton through variousexamples. Let us choose the textbook example, that of the function y = xn, wherethe process the physicist uses is the following: Newton assumes an infinitely smallinterval of time, which he calls o, and he calls the infinitesimal increases in x andy, xo and yo. To find the relation between these two variables, Newton replaces x iny = xn, x with x+ xo and y with y + yo. We thus have:

y + yo = (x+ xo)n

Newton then develops the right-hand side of the expression into an infinite seriesusing the binomial formula. The binomial formula, known at least from Pascal’s time,states that:

(x+ y)n = xn +

%n

1

&xn&1y +

%n

2

&xn&2y2 + ...+

%n

n# 1

&xyn&1 + yn


with:%n

k

&=

n!

k!(n# k)!

On applying this formula to the case under consideration, Newton obtains:

y + yo = xn + noxxn&1 +n(n# 1)

2o2x2xn&2 + ...+ onxn

He then removes y = xn from each side and divides the two sides of the equationby o, obtaining:

y = nxxn# 1 +n(n# 1)

2ox2xn&2 + ...+ on&1xn

He then ignores all terms that still contain the infinitesimal o and he finally arrivesat:

y = nxn&1x

which can again be written as:

y

x= nxn&1

yx being the slope of the curve, the above formula constitutes, in a de facto manner,

the normal formula for what we today call the derivative of the function y = xn. Theinverse operation is then easy to obtain based on this. Given a function y = xn, theprimitive is none other than: xn+1

n+1 .

5.2.2. Leibniz’s contribution

In 1672, Leibniz, employed in the service of the Elector of Mainz, was part ofa diplomatic mission to the court of the King Louis XIV. During his stay in Paris hemade the acquaintance of Ch. Huygens (1629–1692), who introduced him to the worksof Cavalieri, Roberval, Pascal, Descartes, Gregory and Wallis. In 1676, Leibniz leftParis to pursue his political career in the service of the Elector of Hanover. And it wasonly from 1684 onwards, in a series of quite scattered, short articles that appeared inthe Acta Eruditorum (a journal that was founded with his support in 1682 in Leipzig)that he published his essays on differential calculus.

Leibniz’s inspiration was very different from that of Newton. Thirty years later,Leibniz claimed to have drawn his inspiration from a passage in Pascal’s Traité des


sinus du quart de cercle (Treatise on the sines of quarter-circles) on the characteristictriangle. He realized, in fact, that the solution to the classic problem, which consistsof finding the tangent to a curve, depends on the ratio of the differences between theordinates and abscissas when these become infinitely small. In this context, squaring(in other words, computing the area under the curve) depended on the sum of theordinates or infinitely thin rectangles raised on the infinitely small intervals of thex-axis.

Indeed, it was from 1675 that Leibniz began developing his ideas on the question,and he initially did this based on purely combinatorial considerations, following thelogic of his earlier work, notably the De arte Combinatoria in 1666. In particular,Leibniz declared, in 1672, in a letter to Oldenburg, that he had studied the series ofsquares of the type:

0, 1, 4, 9, 16, 25, 36

and observed that the difference between consecutive squares gave the series of oddnumbers:

1, 3, 5, 7, 9, 11

He also remarked that the difference between these differences was constant andequal to two and that, in addition, the sum of the first differences is equal to the lastterm of the series of squares. In this case:

1 + 3 + 5 + 7 + 9 + 11 = 36

These considerations were at the origin of his differential calculus. In effect,Leibniz immediately saw that we can interpret the series of numbers 0, 1, 4, etc., asthe series of values of a function (in the present case, a function of the type y = x2,as it concerns squares).

Thereupon, the difference between two numbers is the difference between twoneighboring values of the function. Leibniz denotes this difference by l. Abbreviatingthe Latin omnia to omn, he then denotes the sum of the differences by:

omn. l = y

However, Leibniz soon preferred dy to l and replaced “omn.” with a stylized signfor sum, a sort of elongated S that remains, even in modern writing, the sign for theintegral –

'. The above relation thus becomes:

(dy = y


These considerations, which were initially purely combinatorial, would beaugmented by a geometric observation. In his first publication on differentialcalculus, the Nova methodus pro maximis et minimis in 1684, Leibniz studied whatwas then called the tangents problem. This problem is, generally, that of calculatingthe abscissa of a point for which we have the ordinate. That is, in the language of the17th Century, the length of its “subtangent” (the segment TP in Figure 5.1).

Figure 5.1. The tangents problem

The tangents problem had been studied by Isaac Barrow (1630–1677), Newton’spredecessor to the Chair of Mathematics at Cambridge university. Barrow considereda parabola of the type:

y2 = px

Then, upon replacing x with x+ e and y with y + a, he obtained the equality:

(y + a)2 = y2 + 2ay + a2 = px+ pe

Then, removing y and px, as well as the term containing a power of a, which is aninfinitesimal product, he obtained:

2ay = pe

Hence:

a

e=

p

2y

Moreover:

a

e=

MP

TP


as the triangles MRN and MPT are similar. As MP is the ordinate of the point M, theproportion:

MP

TP=

p

2y

made it possible to calculate the length of the subtangent TP.

This problem led Leibniz to consider the infinitesimal triangle NMR as acharacteristic element of the curve whose three sides remain perfectly determined, asa result of the relation of similarity with the triangle TNQ, formed by the subtangentTQ, the ordinate QN and the length of the tangent TN.

The result of this is that even if dy and dx are arbitrarily small quantities, theratio dy

dx has a finite value, that is, that of the ratio NQQT . This gave a definition of the

differential: dx being any quantity, the differential dy is defined by:

dy

dx=

y

subtantgent

From this, Leibniz stated the rules to calculate:

1) the differential of a sum: d(x+ y);

2) the differential of a product: d(xy);

3) the differential of a ratio: d(x/y);

4) the differential of an exponent: d(xn).

He thus created, in his own words, a veritable “algebra of infinitesimals” andapplied these rules to research on tangents to a curve, to the computation of theminima and maxima of a function, to the computation of points of inflection, etc.

He later added to the rules for his algebra, formulae to calculate the differentialsof logarithmic and exponential functions; studied the curvature of curves using theosculating circle method; introduced, further, differentials d(n)x of an order greaterthan 1. In short – in accordance with his constant method, Leibniz never stoppedgeneralizing his discovery.

Two characteristics of his calculus set it apart from Newton’s:

1) Leibniz based his calculus on the concept of the differential, which he conceivedof as an indivisible and simple element, as the monads are in philosophy;

2) the computation of differences is, therefore, the fundamental operation of theLeibnizian calculus, summation being the inverse operation. Areas and volumes thus


become infinite sums of infinitesimal elements, but Leibniz directly calculated thevalue of these sums by inverting the operation of derivation, thus stumbling uponthe concept of the definite integral, contrary to Newton who, as we have seen, usedindefinite integrals and calculated areas and volumes using their rate of variation.

5.2.3. The impact of calculus on Leibnizian philosophy

Within Leibnizian philosophy, we can, we believe, identify different traces of hismajor scientific discoveries, notably that of the infinitesimal calculus.

5.2.3.1. Small perceptions and differentials

The first of these can be easily found when reading Nouveaux essais surl’entendement humain (New essays on human understanding), which was a textwritten to counter Locke’s philosophy. This was written in 1704, but only publishedin 1765. One text, which is also very famous, that has proven to be particularlysignificant is the one that speaks of what Leibniz calls “petites perceptions” (smallperceptions) or “insensible perceptions”. That is, perceptions without “aperception”,perceptions that we do not perceive: the continuous sound of a mill-wheel for thosewho live close by; the sound of each wave in the general to-and-fro of the sea; theminuscule sounds that make up, in general, the loud “noise” of the world. As Leibnizhimself said:

“There are hundreds of indications to lead us to the conclusion that atevery moment there is in us an infinity of perceptions, alterations in thesoul itself that we aren’t aware of and don’t reflect on. We aren’t awareof them because these impressions are too tiny and too numerous, or toounvarying. In either case, the perceptions in question when taken singlydon’t stand out enough to be noticed. But when combined with othersthey do have their effect and make themselves felt, at least confusedly,within the whole. That’s how we become so used to the motion of a millor a waterfall, after living beside it for a while, that we don’t attend to it.Its motion does still affect our sense-organs, and somethingcorresponding to that occurs in the soul because of the harmony betweenthe soul and the body; but these impressions in the soul and the body,lacking the appeal of novelty, aren’t forceful enough to attract ourattention and our memory. Attending to something involves memory.Many of our own present perceptions slip by unconsidered and evenunnoticed, but if someone alerts us to them right after they haveoccurred, e.g. making us take note of some noise that we’ve just heard,then we remember it and are aware of having had some sense of it. Thus,we were not aware of these perceptions when they occurred, and webecame aware of them only because we are alerted to them a little,


perhaps a very little, later. To give a clearer idea of these tiny perceptionsthat we can’t pick out from the crowd, I like the example of the roaringnoise of the sea that acts on us when we are standing on the shore. Tohear this noise as we do, we have to hear its parts, that is the noise ofeach wave, although each of these little noises makes itself known onlywhen combined confusedly with all the others, and wouldn’t be noticedif the wavelet that made it happened all by itself. We must be affectedslightly by the motion of this one wavelet, and have some perception ofeach of these noises, however faint they may be. If each of them had noeffect on us, the surf as a whole, a hundred thousand wavelets, wouldhave no effect either, because a hundred thousand nothings cannot makesomething! And here is another point: we always have some feeble andconfused sensation when we are asleep, however soundly; and theloudest noise in the world would never waken us if we didn’t have someperception of its start, which is small, just as the strongest force in theworld would never break a rope unless the least force strained it andstretched it slightly, even though the little lengthening that is produced isimperceptible”.

Why perceptions that we are not aware of? The beginning of the text mentionsthree non-exclusive reasons. It could be that these perceptions are too small, that is,not intense enough for us to attend to them. Or they could be too numerous (in thiscase, they are not “countable”, that is, they cannot be counted using whole numbers.It must be assumed that they exhaust this set). Or, again, they are too closely linkedfor us to be able to distinguish them. In all these cases we return to the grand universalcontinuum, which Leibniz (familiar with this idea of the Ancients, according to whichnatura non fecit saltus) compares to the set of reals.

One of the objections that may be raised against the existence of such perceptionsis that mentioned by Leibniz himself in the second part of his text. Namely, the ideathat if we cannot perceive these perceptions then, in truth, it is possible that they donot really exist: we either believe that no impression of them reaches our senses, or weassume that if there were an impression and movement in the nerves, our nerves didsend out signals. Leibniz, however, refutes this position: there is indeed a movementthat reaches our organs and, by virtue of the correspondence between soul and body,which (using that beautiful Leibnizian expression) inter-express themselves and sowhat happens in the body must necessarily leave its trace on the soul.

What, then, is the solution? It is that these “petites perceptions” are perceptionsthat are dulled by habit, that is, by repetition. Robbed of any novelty, they weakenand no longer claim our attention. They are thus not memorized and appear not tohave existed. This is because, in order for a perception to have an existence, it mustbe momentarily fixed in memory, however briefly this is. But such a process has high


costs, with the result that ordinarily our attention is discontinuous. We need a distinctreminder in order to pay attention to that which we no longer notice.

Then comes the third part of the text and this fabulous example of the sound ofthe sea – composed of a vast number of sounds, each associated with a separate wave.Leibniz makes the following argument: first, if we hear the whole, we must haveheard the parts that make up the whole. If these parts were nothing, we would nothave the whole. A hundred thousand nothings do not make anything. The passagefrom the unconscious to the conscious strongly assumes that the unconscious was nottotal, otherwise we would not be able to explain the change of state. Finally, in thecase of the transition from sleep to wakefulness, we would not be able to explain thephenomenon of waking up. The loudest sound in the world could not wake us unlesswe had already begun to perceive this sound earlier.

As we can see, Leibniz uses differential calculus for psychological purposes inthis text. The “petites perceptions” are, in fact, faint quantities, analogous to thedifferential mathematics he invented. They are, in truth, differentials of conscience,with apperception performing a sort of integration of these differentials. In the caseof the sound of the sea, it is ongoing and continuous. In the case of the noise made bythe mill, it only happens above a certain threshold of awareness or if someoneremarks on it.

These insensible perceptions, which went unstudied by Descartes, are directlyrelated to the model of the infinitesimal calculus that irradiated Liebniz’s philosophythrough these “petites perceptions”:

1) these perceptional differentials would, in effect, bring home the complexity ofour tastes and all images involving tangible qualities that seem clear to us whenassembled but meaningless in their parts: the entire field of feelings, in particular,that Descartes spurned in his philosophy, find their place in Leibniz’s philosophy.Working through mathematics, thus, has proven to be a particularly productivemethod for philosophical thought. The infinitesimal calculus model makes it possibleto include a field that was, until this point, systematically excluded. The concept ofthe union of soul and body (and of all that relates to this) is, for Descartes, a thornyone that is difficult to understand as it does not allow for “clear and distinct” ideasrelated to all concepts that touch upon this union; the philosopher, therefore, couldhave no real knowledge of this phenomenon;

2) the impressions that the objects around us make on us, inasmuch as they are“petites perceptions”, go beyond that which we perceive3. We are, thus, spatially and

3 This does not, of course, refer to the Freudian unconscious (that bubbling cauldron of impulsesand desires) but rather of that which Freud himself would one day call the “preconscious”. That


temporally related to something that lies beyond our ordinary consciousness. That is,our perception of the universe far exceeds our conscious perception of the universeand this holds true for both space and time. We perceive, without realizing it,elements that are beyond those in the here-and-now of our current perception and weare thus virtually connected to the entire universe4. Such that if we were able to takecognizance of these small perceptions, we could widen our perceptions not only inspace but also in time, especially in the future5. There is nothing mystical about this,even if Leibniz believes that God, who sees farther than all humans, can, as a resultof his omniscience, see everything in the universe;

3) we ourselves are, of course, not God. However, by applying this differentialcalculus algorithm to psychology, we can try to move from the local to the global andthus, at least rediscover long-buried memories. This would only require concentratingour attention and our memory on some fact, a bit like developing a function in thevicinity of certain points;

4) just as the transition from sleep to wakefulness assumes a continuum, so doesthe transition from life to death, or, in the other direction, from non-life to life.Consequently, nothing is ever completely dead: there are, in death, the differentialsof life. Death is, at the heart of it, nothing but a faint or suspended life. This wouldhint, though Leibniz does not go so far as to say this, that the operation might bereversible;

5) the fifth consequence of the existence of insensible perceptions, related toinfinitesimal consequences, concerts the typically Leibnizian doctrine of thepre-established harmony of the soul and the body. Leibniz believed that the soul andbody are as two clocks which have, once and for all, been perfectly synchronized toindicate the same time down to the last second. If we move away from the clockmetaphor, which is quite inadequate, it must be said that the continuum of movementin the body and the continuum of perceptions of the soul correspond exactly, a littlelike two differential varieties that are related by a diffeomorphism;

6) the importance of the infinitesimal calculus can also be measured in the fieldof practical philosophy, especially with respect to the question of liberty. Leibniz, theinventor of the calculus, was no more able than Descartes to believe in the existence ofa free will – an ability to choose that could eventually manifests itself in an arbitrary

is, a set of elements that are not currently present in consciousness but which can easily bebrought into consciousness if we attend to them.4 In Monadology (ß60–61), Leibniz also goes back to the old adage by Hippocrates: “sumpnoapanta” – everything breathes together.5 E. Jünger makes excellent use of this possibility in his book On the Marble Cliffs, where anold woman, massaging the body of a young man who is off to war, believes that she can feel thewound he will receive. See the commentary in [PAR 91].


manner or takes the form of freedom of indifference, lower than freedom, according toDescartes, but a human attitude that is always possible. For Leibniz, on the contrary,such a situation is precluded, and the problem of Buridan’s ass (an ass that is equallyhungry and thirsty placed at an equal distance from a pail of water and a stack ofhay) is a false problem. In reality, insensible perceptions also correspond to insensiblevolitions. And if we decide on one path rather than another, it is because we have beenpushed to choose it, even unbeknownst to us;

7) the insensible or “differential” perceptions are also the source of our disquiet:disquiet (in Latin: in-quietas, un-rest), is a sort of differential of agitation that createsuneasiness or pain. These are, of course, limited to the psychological aspect, reducedin intensity as they are often on the threshold of being sensible, and the disquiet mayalso have positive aspects: sharpening our desire or insensibly enhancing pleasure.We can see from this that the infinitesimal calculus makes it possible to extend theancient Stoic calculus of pleasures, which was, even in that time, a rational calculusthat arrived at the boundary of sensitivity, almost going over into insensibleness,allowing us, in brief, to extract the maximum possible pleasure from life;

8) according to Leibniz, these “petites perceptions” could also lead to synesthesia,these remarkable connections that people believe can be made between colors, soundsand other tangible qualities that Rimbaud’s poems would, one day, speak of and thatthe “mouth organs” of Des Esseintes would bring into play in Huysmans’ A rebours,or again, which can be seen in Boris Vian’s “pianococktail” in his book L’Ecumedes jours (Froth on the daydream, in English). In short, the infinitesimal calculus hadinnumerable consequences on how we interpret tangible aspects of our life, but theyalso go far beyond this.

5.2.3.2. Matter and living beings

For Leibniz as for Descartes, matter was a geometric field made up of figures andmovement. In this sense, it was infinitely divisible. However, this infinite divisibilityhad a different meaning for Leibniz than the Cartesian sense, which stopped withrevisiting the Aristotelian concept of potentially infinite divisibility. Monadology(section 65), on the contrary, presents a very different conception of matter:

“Every portion of matter is not only divisible to infinity, as the ancientsrealized, but is actually sub-divided without end, every part divided intosmaller parts, each one of which has some motion of its own, rather thanhaving only such motion as it gets from the motion of some larger lump ofwhich it is a part. Without this infinite dividedness it would be impossiblefor each portion of matter to express the whole universe”.

The Leibnizian expression “actually sub-divided without end” signifies thatmatter is infinitely divided and sub-divided in actuality and not potentially. In otherwords, each portion of the material field is, to infinity, as concerns the figure, an


actual infinitesimal, that is the infinitesimal of the differential. Similarly, each portionof the movement that animates this matter is, to infinity, an infinitesimal (ofmovement) in actuality. If this were not so, Leibniz explains, the generalinter-expression of substances (that is, the theory by which each part of the real, eachportion of matter, expresses the whole of the universe) would be impossible. It is onlybecause the whole is nothing but an extension of the singular, because what manifestsitself on a grand scale can only be conceived of as the extension of that which iscontained on a smaller scale (just as a curve may be reconstituted based on itsconsideration over a very small interval), that Leibniz can justify such acorrespondence.

The conception of living beings as infinite machines (Monadology, section 64)comes from this. Unlike mechanical machines that are finite machines (for examplethe cog on a brass wheel, which is not itself a wheel and whose parts are not themselvescogs), Leibniz states that living beings are “machines even in their smallest part”.We have here another consequence of the infinite division of matter in actualityand this vision of the world would, three centuries down the line, have two notableconsequences:

1) that which we call (dating from B. Mandelbrot’s work in the 1970s) “fractalstructures” (structures whose parts are identical to the whole on the smallest scaleand, in principle, infinitely) are in line with the Leibnizian vision that anticipated thisinternal homothety (or fractal dimension) of certain natural or artificial objects;

2) In sync with this, the nanotechnology project based on the idea of the physicistRichard Feynman, according to whom “There is plenty of room at the bottom” is alsoa kind of proof for Leibnizian ideas. The idea of using sub-atomic space to constructminuscule machines which would then themselves construct even smaller machines,and so on, until Planck’s limit is reached, has its source in this Leibnizian concept ofmatter. This is, in itself, linked to his interpretation of the infinitesimal calculus and tothe notion of the infinitesimal in actuality.

5.2.3.3. The image of order

An article in Leibniz’s Discourse on Metaphysics (1686, article VI), affirms againthat “God does nothing that isn’t orderly” and this idea that the whole of the universeis orderly and that there cannot, therefore, be any disorder in the world, is explainedby Leibniz using arguments that are, again, borrowed from the mathematics that heinvented:

“Suppose that someone haphazardly draws points on a page, like peoplewho practice the ridiculous art of fortune-telling through geometricalfigures. I say that it is possible to find a single formula that generates ageometrical line passing through all those points in the order in whichthey were drawn. And if someone drew a continuous line which was now


straight, now circular, now of some other kind6, it would be possible tofind a notion or rule or equation that would generate it. The contours ofanyone’s face could be traced by a single geometrical line governed by aformula. But when a rule is very complex, what fits it is seen as irregular.So one can say that no matter how God had created the world, it wouldhave been regular and in some general order. But God chose the mostperfect order, that is the order that is at once simplest in general rulesand richest in phenomena, as would be a geometrical line whoseconstruction was easy yet whose properties and effects were veryadmirable and very far-reaching”.

The five fundamental points of this text are:

1) a curve can always connect points placed randomly, in the order that they wereplaced;

2) several successive curves (line, circle, etc.), of different forms, may in realitybe only one curve, and be expressed by the same equation;

3) all natural forms (a face, for instance) are the realization of some geometriccurve that is more or less complex. The result of this is that with those curves thatare able to take any concrete morphology, there is no longer any opposition betweengeometry and finesse. Geometry was able to perfect itself through the invention of theinfinitesimal calculus, and today encompasses the smallest sinuosity of the real world;

4) there is nothing in the universe that cannot be expressed as some curve, howevercomplicated (irregularity is always underpinned by order and regularity);

5) perfect curves illustrate the simplest hypotheses related to the richest effects.

6 In June 1686, Leibniz published his Meditatio nova de natura anguli contactus et osculi, inwhich he introduced the idea of an osculating circle. He demonstrated that the consideration ofinfinitely small parts of a curve made it possible to not only study its direction, that is, its slope orincline, but also made it possible to study the variations of this direction, that is, its curvature.And as the direction of curves is measured by the simplest line, the tangent, the curvature isalso measured by the simplest curve, the circle that, for a given point, the same direction andcurvature has the minimal angle of contact between itself and the curve. This is the angle ofosculation, being the smallest possible value. The mathematician then generalized this conceptof the osculating circle to the osculating curve, showing that known curves could be used todefine others. To do this, it was enough to examine at how many points they intersected eachother (see Acta Eruditorum, June 1686, in [LEI 95, p. 124]). We can thus understand that acurve may be successively composed of lines, circles and other curves of any kind, all of whichfollow its direction and its curvature and all points.


For a greater understanding of this text, which has largely been ignored bycommentators7, it would be useful to study similar texts. We find an early variant,two years later, in the opuscule that is usually called Specimen Inventorum (1688)[LEI 96, p. 293], in which Leibniz returns to the same subject:

“...just as it is impossible to draw any line which is not geometric,however unsteady the hand, and which does not have a constant anddecisive nature common to all of its points, there is no series of things,nor any imaginable manner of creating the world so chaotic that they arenot based upon some proper order, fixed and determined, and on the lawsof progression, even though, as with lines, certain series have greaterpotential and simplicity than the others and, consequently, greaterperfection and achieve more using less”.

As we can easily see, if we carry out the exercise proposed by Leibniz (tracingpoints at random and observing that it is always possible to connect them all througha curve), the problem, without adding any more conditions, is completelyunder-determined. In fact, if we consider a series of points randomly placed in aplane, it is possible to connect them all not just through one curve but an infinity ofcurves. Leibniz’s preference (the principle of the best) is evidently for the curves thathe mentions in the earlier text, that is, the simplest to hypothesize and the richest ineffects (cycloid, spiral, catenary, tractrix, etc.) which are, every time, optimizationcurves. But this is not the nature of all curves and, using points scattered at random, itis quite rarely possible to construct a curve of this kind. How, then, do we resolve theproblem of choosing the curve? Leibniz’s response, offered in his Letter to Varignon(1707), is that the solution can be arrived at through a procedure that we today callinterpolation:

“To explain myself in the style of algebra, I hold that if, imitating Mr.Hudde – who claimed to be able to assign an algebraic curve whosecontours would match those of a known face – we could express,through a formula of a higher characteristic, some essential property ofthe universe, we would be able to read from this what the successivestates of all of its parts would be across all of the assigned times. And

7 Only Michel Serres has referred to this text in his Eloge de la philosophie en langue française(Eulogy of philosophy in French), but has also spoken very little about it [SER 85, p. 225and the following]. He stops with showing that Leibniz originated procedural mathematics(infinitesimal calculus, combinatorial, theory of coding, etc.) whose models make it possibleto construct objects step-by-step. But Serres does not specify what procedure he means in thatprecise case.


thus it happens that we cannot find even a single natural event thatdisproves this grand principle; on the contrary, everything that we knowprecisely, justifies this perfectly...”.

Leibniz – and later, Laplace – undoubtedly held several illusions as to the powerof analysis to explain the causal laws of the universe (the existence of chaoticphenomena related to the sensitivity of dynamic systems to initial conditions does,evidently, escape him). But what he finds important is expressed in the next part ofthe text: “There may be continuity in things that exist simultaneously, even thoughthe imagination only sees discontinuous jumps: this is because many things appear,to our eyes, to be completely disparate and disunited but which we find are,nonetheless, perfectly similar and united within themselves, if we are able to knowthem distinctly. If we only considered the external configuration of parabolas,ellipses and hyperbolas, we would be tempted to believe that there is an immense gapbetween each of these kinds of curves. However, we know that they are intimatelylinked such that it is impossible to place any other intermediary between two, thatwould help us to move from one to the other with even more imperceptible nuances.

“I believe, therefore, that I have good reason to believe that all of the differentclasses of beings that, together, form the universe are, in God’s idea (and God isdistinctly aware of their essential gradations) just so many ordinates of the samecurve whose union cannot tolerate others between two ordinates, as this wouldindicate disorder and imperfection” [LEI 72, p. 376].

In this text, the idea is still the same: that of a regularity hidden under apparentdisorder, or of a continuity underlying the apparent discontinuities of the universe; theidea of an absolute order, comparable to the ordinates of a curve, being associated herewith the “gradations” between different beings such that no interpolation is possiblebetween them.

As we have seen, the problem that Leibniz poses in the first two texts was aninterpolation problem. Interpolation is the process consisting of interposing one ormore terms determined using calculus in a series of known values. Arising in the17th Century with Gregory8 [GRE 39] and Newton9, the interpolation problem then

8 It is, of course, quite unlikely that Leibniz knew of this letter. Nonetheless, certain specificcases from Gregory’s formula had been published, a few decades earlier, by Briggs. Accordingto E. Meijering [MEI 02], a historian of interpolation, Briggs had already described, inhis major works [BRI 24, BRI 33], rules and examples of interpolation and we also know[GOL 77, LOH 65] that from 1611 onwards, Harriott used equivalent formulae.9 We know of several contributions that Newton made on this topic. Liebniz, however, couldnot have known of these: a letter to Smith in 1675 and a letter to Oldenburg in 1676. See the


developed in two forms: as polynomial interpolation under Lagrange and Hermitte,and then as a theory of spline functions in the 20th Century.

Polynomial interpolation (to which Leibniz alludes) responds precisely to thefollowing problem: assume that we have measured a certain line at the equidistantpoints x0, x1, x2, etc. and that we wish to obtain its value at all of the intermediarypoints. Gregory (Letter to Collins, 1670) and then Newton (Principia Mathematica,Book III, lemma V, 1687), demonstrated that it is always possible to find apolynomial function that passes through all of these points and thus reveals the valueof all of the intermediary points.

In the earlier formulation, the interval is assumed to be divided into equidistantsegments. In other words, the “step” (that is, the length of each segment) is constant.And this result can, indeed, be generalized to the case where the interval is dividedinto any segments and where, this time, the “step” is variable.

The subsequent generalization of Leibniz’s problem then consists of asking if itwould be possible to find a curve that passes through an infinite number of pointsdrawn at random. The situation is the same as refining the subdivisions of the intervalover which we seek to reconstitute the curve and, therefore, the same as having thelength of these subdivisions tending to zero. The step, at this stage, becomesinfinitesimal and it cannot be reduced any further (which would correspond, more orless, to the situation evoked by Leibniz in his Letter to Varignon, if we admit that thecontinuum of the real is a divine creation). We thus seek, definitively, in this case, toapproximate a curve with the help of another curve, which may be a polynomialfunction and which we call a Lagrange interpolation polynomal.

The most general case we can approach today is that where a function f , to beapproximated, is a function of complex value. We thus take a subdivision series Sn

and a Lagrange interpolation polynomial Ln(f), and we study the convergence ofthis polynomial Ln(f) toward the function f . We then see that it is not at all evidentthat this is a guaranteed convergence. This requires certain additional conditions (inparticular, the derivability of f in an open disk containing the considered interval).

Even in the case of the real, there are many problems that arise. In the case of abanal, real analytic function and a subdivision of the interval with constant step, ithappens that the approximation polynomial does not converge on f , in the sense that

manuscript titled Methodus Differentialis, which was published only in 1711 [TUR 60]. Alsosee: [JON 11, pp. 93–101], [WHI 81, pp. 236–257], [FRA 27], [NEW 87, Book III, lemma V],and, finally, a manuscript titled Regulae Differentiarum, written in 1676 but found and publishedonly in the 20th Century.


it can, for example, correctly approximate the function in the vicinity of zero, but itmoves to the edges of the interval (the Runge phenomenon).

The solution to these problems leads to the theory of “spline functions”, whichrequires additional hypotheses concerning the ! functions that should approximatethe function f . Leibniz specified neither the degree nor the differentiability of suchfunctions. In these conditions, and contrary to his declaration, it is not always possibleto pass a curve through points laid out beforehand, which would tend to prove thatthere are many more things “out of order” in this world than he believed.

In order to resolve all these equations, the ! functions must be:

1) polynomials of degree ' n;

2) n# 1 times differentiable over the considered interval.

These conditions impose a certain regularity on the functions. They define the classof spline functions (a kind of flexible rod). However, they are not sufficient:

3) for each division of the interval, we must also have the equality !(xj) = f(xj).

This condition prevents phenomena such as the “Runge” phenomenon.

However, in such a case there are still n # 1 parameters free if the function is ofdegree n. We will thus still have a vast number of possible curves and it would then berational to take the simplest curves. The case where n = 2, however, gives a functionthat is too rigid. The case where n = 3 (cubic spline functions), on the other hand, isa good candidate as it makes it possible to impose the condition;

4) !!(a) = f !(a) and !!(b) = f !(b)

It can then be demonstrated that the four conditions (1)–(4) determine the curve! approaching the series of points f in a unique manner. This function is denoted bySp(f) and is called the cubic spline function interpolating f of the order p.

We thus obtain the desired result. It must, however, be noted that the explicitcalculation of this function often requires the resolution of a system of linearequations that are quite complicated. The situation could be worse for splines of oddorders greater than three. We thus limit ourselves to cubic splines10.

In conclusion, we see that modernity has transformed the Leibnizian problem asthere is no longer any need for external considerations or to distinguish between

10 For more on these mathematical developments, the reader is invited to consult: [DAV 75,STE 81, ZAM 85].


possible approximations. There is no need for an “at best”; no condition concerningthe universe as a whole is necessary any longer in this procedure that onlydemonstrates that the most regular curve must satisfy the axiomatic of splinefunctions of an odd order greater than or equal to three.

However, in order to obtain such a result, it was necessary to impose additionalnorms on the functions that satisfied the initial criterion (passing through ordinatepoints distributed randomly), ensuring:

1) sufficient conditions of regularity;

2) a coincidence of two functions at the points of subdivision of the interval;

3) a perfect adherence to the limits.

These rules, while not being as exigent as the Leibnizian hypotheses, translatewhat Leibniz expressed in more metaphysical term in a technical manner.

In other words, in order to ensure regularity we introduce it as a norm in thehypotheses that the functions must satisfy. The world, thus, appears ordered becausewe have axiomatically chosen the means to verify that it is. There is no real reason,otherwise, why this should be so!

It remains that Leibniz made good use of the possibilities that infinitesimalcalculus offered him (even if this was done a little too ambitiously and, thereupon,fallaciously) in order to justify a philosophical view of the world around which heconstructed a veritable philosophy of order (a contestable one, as we have seen).

As C. Houzel notes, there is thus no doubt that the discovery of the infinitesimalcalculus had a number of impacts on Leibnizian philosophy.

Finally, we can note in particular that the introduction of the idea of infinite seriesin philosophy had considerable impact.

On the one hand, as concerns a decreasing series based on the universal law ofcontinuity, it tends to make the final term of the series appear as a relative, rather thanan absolute, zero. This bolsters up the spiritualist idea that when the sensible realityof an object fades away, what remains is not then nothingness, but an infinitesimalessence of the object in question, which thus survives the material.

On the other hand, the fact that God is Himself situated at the infinity of causes andeffects makes it possible to break free of the idea of predestination, which has alwaysbeen seen as fettering human freedom. Indeed, C. Houzel writes, “at infinity, the seriesof freedoms take the form of a spiritual destiny, and the infinite, therefore, transmutesall: the infinite constraint into a finite liberty and, conversely, the moral of good and


bad, on a finite scale, in the logic of the best of all possible worlds” [HOU 76, pp. 79–80]. We can here refer to the important texts in Theodicy (I, 20–26), and also to thisspecific passage that will, one final time, evoke the problem from which we startedoff:

“It should be no cause for astonishment that I endeavor to elucidate thesethings by comparisons taken from pure mathematics, where everythingproceeds in order, and where it is possible to fathom them by a closecontemplation which grants us an enjoyment, so to speak, of the visionof the ideas of God. One may propose a succession or series of numbersperfectly irregular to all appearance, where the numbers increase anddiminish variably without the emergence of any order; and yet he or shewho knows the key to the formula, and who understands the origin andthe structure of this succession of numbers, will be able to give a rulewhich, being properly understood, will show that the series is perfectlyregular, and that it even has excellent properties. One may make this stillmore evident in lines. A line may have twists and turns, ups and downs,points of reflection and points of inflection, interruptions and othervariations, so that one sees neither rhyme nor reason therein, especiallywhen taking into account only a portion of the line; and yet it may bethat one can give its equation and construction, wherein a geometricianwould find the reason and the fittingness of all these so-calledirregularities. That is how we must look upon the irregularitiesconstituted by monstrosities and other so-called defects in the universe”[LEI 69, p. 263].

As God is Himself a mathematician, it is essential to rise to this level, in order toperceive the world clearly. In other words, it is essential to become, ourselves,mathematicians, At this level, irregularities and disorder disappear and all shouldcome into order. Modernity, of course, soon lost this optimism. From the time ofGregory Chaitin’s work, we not only know that there exist incompressible series (andfor which the only known algorithm is to display them) but also that only a smallnumber of mathematical problems can be resolved mathematically. It must, thus, beassumed either that the other problems arise from a super-mathematics that isunknown to us, or that God is not a mathematician, or, again, that there is notranscendental order and that relative chaos reigns supreme almost everywhere in theworld.

5.2.4. The epistemological problem

A final topic for reflection is the epistemological problem posed by theinfinitesimal calculus: we will, here, only ask two questions of a purely mathematical


nature, that greatly troubled 17th Century minds with respect to this problem that wasrich in possibilities but also worrying:

1) the first question is that of the validity of calculus. We saw this, in particular, inNewton’s and Barrow’s examples – when we use the differential calculus, we neglectinfinitesimal terms and products. This same problem, of course, also arises withLeibniz’s work when we identify the derivative with respect to the ordinate on thesub-tangent. In all cases, we can identify at most a portion of the curve and a portionof the line and we neglect infinitesimals;

2) the second question is: what are, veritably, infinitesimal elements? Cavialierisaid that they were “indivisibles”, Newton called them “fluxions”, i.e. theinstantaneous rate of change of a quantity. Leibniz himself responded with“infinitesimally small in actuality”. This response, however, presupposesdevelopments that do not follow from it.

The solution that mathematicians would find, eventually consisted of eliminatingthe concept of “infinitesimal” from mathematical language and giving calculus afinitist interpretation by specifying the concept of “limit”. As Léon Brunschwicgdemonstrated in Steps in Mathematical Philosophy, this solution began to appearwith Lagrange, who introduced the concepts of derivatives (first, second, etc.) in thecontext of a theory to develop a series of functions, that is, a purely algebraic manner,close to that of Newton [BRU 81, pp. 242–249].

Unfortunately, Lagrange – as was already the case with Newton – continued topresuppose the convergence of this series (something that would only be establishedmuch later by Cauchy and Abel) such that the justification for the infinitesimalcalculation would, even in the 18th Century, only be of a pragmatic sort.

Toward the end of this century, Lazare Carnot, in his Refections on the InfinitesimalCalculus (1797), would continue to cite, as the only justification for calculus, the factthat, using calculus, errors corrected themselves as the algorithm was supposed toautomatically correct false hypotheses that were introduced through approximations,given that we are then led to neglect quantities of the same order [CAR 21, pp. 39–40; GIL 79, pp. 160–161]. The true justification would only be seen in Cauchy’s work.Cauchy precisely defined the concept of limit, in a manner that was totally independentof geometric intuition, as he defined it in arithmetic terms. He wrote in his Course inAnalysis in 1821 at the Ecole Polytechnic, “When the values attributed to a variableindefinitely approach a fixed value until, ultimately, they differ from this in as small aquantity as we wish, this quantity is called the limit of all the others” [CAU 97, BOY49, pp. 272–273]. He also proposed the interpretation of the derivative that we knoweven today, that is, the derivative as the limit of the ratio of the increase in the functionto the increase in the variable, this limit l being defined by the fact that the absolutevalue of the difference |an # l| can always be made smaller than a given number %,however small this may be.


The problem that remains is therefore to find out, in the opposite operation, whenwe sum the differentials (that is, when we sum an infinity of elements to obtain anintegral) what this infinite set is that we suddenly have, so to speak, in actuality and towhich we will also attribute a finite measurement.

What is involved here is the difference between the concept of the potential of aset and the notion of measurement11. This difference was not clearly understood bythe thinkers of the 17th Century, as can be seen, for instance, in Spinoza’s confusingtext in the famous Letter XII to Louis Meyer, when the philosopher reflected on thesum of the inequalities of distances between two circles inscribed one within the other;this sum was, he said, superior to any given number, while recognizing that the spacecontained within two circles is necessarily finite.

In Figure 5.2, the sum)

|AB # CD| is, as a Riemann’s sum, a set with infinitepotentiality and, as a measurement, a finite measure – even null – as can be proven bya simple calculation. In effect, if we posit:

Max|AB # CD| = 1 and Min|AB # CD| = 0

the sum of the inequalities of distance, a function of the central angle $ of the largercircle, is identical to the integral of a circular function. This, as we can see from Figure5.2, has a value of 0 for $ = 0 and 1 for $ = "/2. This is thus a sinusoid of which theintegral can be easily calculated in the different quadrants. We obtain:

( !/2

0sin $d$ +

( !

!/2sin $d$ +

( 3!/2

!sin $/d$ +

( 4!

3!/2sin $d$

= (#cos "/2 + cos 0) + (#cos" + cos "/2) + (#cos 3"/2 + cos")

+(#cos 4" + cos 3"/2) = 1 + 1 +#1# 1 = 0

As concerns the other question, namely, the nature of differentials, the definitiveresponse is also yet to be found.

In line with Newton, Lagrange and Cauchy, we simply turn them into elements ofa ratio which is explained through a finitist interpretation thanks to the idea of limit. If,on the contrary, we retain the Leibnizian interpretation of these differentials, that is,if we consider them to be actual infinitesimals, then in order to be rigorous we mustinvent a new set that makes it possible to consider them as numbers in themselves, tocompare them and to homogenize them with the other quantities in the calculation.

11 A set may have an infinite potential but a finite measure. But, in fact, this concept of the“potential of a set” would only be introduced in mathematics by Cantor toward the end of the19th Century. Thus, mathematics would have to wait until 1880 for this clarification.


A

B

C

D

Figure 5.2. The problem of Spinoza’s (Letter XII to Louis Meyer)

In effect, if the expression dx is an infinitesimal, we cannot, in principle, comparex and dx. Comparing two quantities a and b in mathematics assumes that we candetermine whether a is smaller than or equal to b or whether b is smaller than or equalto a.

Moreover, an axiom posited by Archimedes states that if a < b, then there isalways an integer n such that na > b. In the case of x and dx, it is tempting to saythat dx < x. However, we cannot find any number such that n.dx > x, because dx isan infinitesimal and the product of an infinitesimal with any given number remains aninfinitesimal.

In reality, according a rigorous mathematical status to infinitely small or largenumbers (the problem, in this case, is symmetric) assumes an extension of arithmetic,that is, non Archimedian arithmetic defined over a field that is richer than simply afield of real elements.

This analysis was only developed toward 1965 by Abraham Robinson, under thename “non-standard analysis” [ROB 66]. Defined over an extension of R, the R'

(which, in addition to “ordinary” numbers, includes infinitesimals as well as theirinverse, the infinitely large quantities), this new arithmetic makes it possible to extendLeibnizian intuitions and render them coherent and rigorous12.

12 A. Robinson retained the term “monad” to designate the set of non-standard real elementssurrounding a standard real element.

6

Complexes, Logarithms and Exponentials

The objective of this chapter is to show how three chapters of mathematics, whichare all three seemingly in such stark contrast (transcendental numbers such as ", thatwe have already encountered; imaginary numbers, whose introductions we willexplain; and finally, reflections on logarithms and exponentials, through theirdevelopment into series), converged and were discovered to be interrelated.

Several philosophical consequences can be drawn from these analyses. Let us statethese right at the outset:

1) the first of these is that, through the invention of new correspondences whichcannot be deducted from the strict definition of the conditions laid down for a problem,advanced mathematics fell out of the scope of analytic thought and – as Hegel rightlyobserved in a highly important text in Science and Logic – rose up to synthetic aspects;

2) the second is that new mathematical concepts, related to the history anddevelopment of new techniques, finally allows approximations that would never havebeen attempted had it not been for these concepts. The introduction of complexes,logarithms and exponentials will not only bring together three seemingly unrelatedmathematical constants (", e and i), carrying out the first “unification” of completelydisparate domains. It will also result in a new vision of the world, one that has beencontested and debated from its introduction – a fluid universe where solid matter seemsto be practically absent. We will, naturally, be discussing the two revolutions, whichare loaded with implications for philosophy;

3) the last consequence is that the series developments, associated moreover withperfecting the infinitesimal calculus, would suggest a new vision of the world, onethat is more dynamic than the earlier ones, and where the Absolute of the philosopherswould be redefined.



We will attempt to demonstrate these assertions by tracing the course of historyfrom the discovery of " to the introduction of imaginary numbers, and from hereonto the concept of exponentials. We will then present Euler’s elegant formula thatsummarizes the relations between these three key constants in mathematics ( ", i ande) which seemed, initially, impossible bring together. This formula resulted, in a way,in the first “unification” in the field of mathematics. Finally, we will conclude thischapter with an overview of the work of the Polish mathematician and philosopherHoëné-Wronski, a specialist of series and determinants.

6.1. The road to complex numbers

In 1484, in a manuscript titled Triparty, Nicolas Chuquet, a French doctor in Lyonattempted to resolve the equation:

4 + x2 = 3x

That is:

x2 # 3x+ 4 = 0

As we can see, the discriminant of this equation, ! = b2#4ac = 9#16 = #7, isnegative. This eliminates all possibility of discovering real roots. Chuquet noted thisin his own way and things remained here.

A little under a century later, in 1545, a certain Cardan, who possessed a brilliantand innovative mind, asked the following question in Chapter X of his Ars Magna: didthe resolution of certain mathematical problems justify the use of negative hypotheses?He also posed the following problem, that of “Dividing the number 10 into two equalparts whose product is 40.”

Cardan observed that it was in fact possible to resolve a particularly thornyproblem of this sort, by allowing oneself the possibility of using negative roots. Ineffect, if we pose:

10 = 5 +"#15 + 5#

"#15

Then the product obtained is:

(5 +"#15)(5#

"#15) = 25 + 15 = 40.

Cardan’s problem may seem artificial. But through Descartes it would soon findgeometric significance. For example, if we wanted to find the intersection of the line

Complexes, Logarithms and Exponentials 123

D(x, y), whose equation was x + y = 10, with the hyperbola xy = 40, the system tobe resolved is:

x+ y = 10 xy = 40

which gives:

x = 10# y

hence:

(10# y)y = 40

or again:

y2 # 10y + 40 = 0

We have:

& = 25# 40 = #15

and hence the roots are:

y = 5±"#15 and x = 5±

"#15

These Chuquet–Cardan numbers, which are the roots of negative numbers, weregiven the name “imaginary numbers” by Descartes in Book III of his Geometry, whileGauss, in 1831, called them “complex numbers”.

As we have seen, Descartes did not use these numbers. However, they were usedby mathematicians. Bombelli, a disciple of Cardan, also proposed the equivalent ofequalities of the kind:

i2 = #1 or1

i= #i

The notation i ="#1 was, in fact, only introduced in 1847 by Gauss.

What does this number i correspond to? And, in general, what do complexnumbers correspond to?


A complex number is defined as a couple of real numbers: z = (x, y). If z = (x, y)and z! = (x!, y!), then the following expressions are called, respectively, the sum andthe product of two complex numbers:

z + z! = (x+ x!, y + y!) zz! = (xx! # yy!, xy! + x!y)

It is, therefore, evident that complex numbers of the form (x, 0) can be identifiedwith real numbers (in geometrical terms, they are all located along the same line).

On the other hand, the number i corresponds to the couple (0,1). Applying the lawsof multiplication, we can then verify that we have:

i2 = (0! 0# 1! 1, 0 + 0) = (#1, 0) = #1

We thus demonstrate that any complex number z = (x, y) can also be representedin the form:

z = x+ iy

In effect, the complex number z = (x, y) may be represented in a plane, thecomplex plane, whose unit vectors are the vectors (1, 0) along the X-axis and thevector (0, 1) = i along the Y-axis. We call this form of representation the “Cartesianform of complex numbers”.

In this form of writing, the usual laws of elementary calculus (listed earlier) stillhold good and take into account that i2 = #1, we then obtain, for instance:

zz! = (x+ iy)(x!,+iy!) = xx! + ixy! + ix!y + i2yy! = xx! # yy! + i(xy! + x!y)

which brings us back to the formula for the above-mentioned product. The square ofnon-null complex numbers of the form (iy, y) ( R is the negative real number #y2.For this reason, they are called “pure imaginary numbers” and the axis Oy in thecomplex plane is called the imaginary axis.

The complex number z = x# iy is called the “conjugate” of the complex numberz = x+ iy.

We have the relations:

z + z! = z + z! zz! = zz! ¯z = z

It follows from this that if z $= 0, then z $= 0 and 1z = 1

z .


For any complex number z, the product N(z) = zz = x2 + y2 is a positivereal number: it is the square of the distance from the image of z to the origin of thecoordinates.

The positive real number:

|z| =!x2 + y2 =

"zz

is said to be the modulus of z.

The modulus of complex numbers has the same properties as the absolute value ofreal numbers. We have, in particular:

|z| = 0 if and only if z = 0

We also have:

|z + z!| ' |z|+ |z!| |zz!| = |z||z!|

Let us also observe that the inverse of a non-null complex number is equal to1z = z/|z|2. It follows from this that for complex numbers with a modulus 1, i.e.:

|z| =!x2 + y2 =

"zz = 1

their inverse form is the same as their conjugate. In effect, if"zz = 1, z = 1 and

z = 1z .

In this chapter, we will demonstrate the purpose of complex numbers. But beforethat, we must first introduce certain additional ideas concerning the extension ofcalculuses.

6.2. Logarithms and exponentials

Denoting the “square of a” by a!a or a2, “the cube of a” by a!a!a or a3, visiblyhave their origin in the representation of areas and volumes. An evident generalizationleads to the formula a! a! ...! a = an, with n being an integer.

However, from the 14th Century onward, with perhaps the earliest example beingfrom Nicholas Oresme (1323–1382), in his work Algorismus Proportionum (publishedaround 1350), there arose the possibility of having rational exponents (of the type p

q )and not only whole numbers. Oresme proposed formulas of the type:

apq = q

"ap


In addition, the laws of exponentiation are such that ap ! aq = ap+q . Here, weobserve the appearance of a very interesting relation between addition andmultiplication that would be deepened by the developments in civilization.

In effect, the densification of commerce and trade, new banking technologies tomake money, as well as the complexification of the calculations for practical uses suchas navigating the sea or mapping the sky (e.g. Tycho Brahe’s astronomical calculations(1546–1601)) would require simplifications.

Let us take the example of the calculation of interest rates. A capital a = 1 investedfor 1 year at a rate of interest ' becomes:

a = 1 + '(1) with '(1) = '

When invested for 2 years, it becomes:

1 + ' + '(1 + ') = (1 + ')(1 + ') = (1 + ')2

And so, at the end of x years we would have:

(1 + ')x = ax = c

We thus have a geometric progression of common ratio a.

But the banker’s or saver’s problem is often the inverse: given the capital weinitially possess and which we would have at the end of a year, we must calculate thelength of time over which this capital would obtain a given final sum. The question,then, is to find x, if we have a and c.

This operation, which makes it possible to obtain x from a and c, is the inverseoperation of exponentiation. We call this “logarithm” and posit (in modern notation):

x = logac

which states that “x = the logarithm of base a of c”.

Logarithms were invented by the English mathematician John Napier (or Neper)(1550–1617). In 1614, he published a book titled Mirifici logarithmorum canonisdescriptio, which was followed a few years later and independently, by a similartreatise written by the Swiss-born mathematician from Prague, Jobst Bürgi.

Exponentiation is the function which, for x, corresponds to ax; logarithm is theinverse function, which for ax corresponds to:

logaax = x


Logarithms thus reveal a particularly interesting property. As:

logaaxay = logaa

x+y = x+ y

and:

logaax = x logaa

y = y

it follows that:

logaaxay = logaa

x + logaay

And, in general:

lognab = logna+ lognb

We consequently also have:

lognax = xlogna

A particularly convenient base would be base e. We posit, in effect:

logeex = x

We traditionally omit the base for the logarithms of base e (often denoted inabridged form, ln).

Logarithms would, thereupon, be used constantly in calculations and lead to theconstructions of logarithmic tables. Indeed, it is clear that if we know the logarithm oftwo numbers a and b, as well as the logarithm of their product ab, it becomes possibleto replace any multiplication by a simple addition of logarithms. We thus go from logab to ab, without having to multiply a by b, which was an extremely high-performingalgorithm at a time when there were no computers.

The exponential function, or the inverse function of the logarithmic function,possesses similar properties to this. In particular, from the law of addition of powerswe have:

ex.ey = ex+y


6.3. De Moivre’s and Euler’s formulas

Since antiquity, we have seen, it has been known that the characteristic ratiosbetween the sides of a right triangle when one of its angles (let us call it $) is fixed.These ratios are called cos $, sin $, tg$ and cotg$ .

Based on Gauss and Argand’s work, we can then represent complex numbers usinga complex plane (X,O,Y) as shown in Figure 6.1:

Figure 6.1. Argand’s plane

Let us posit OM = #. We can express the real and imaginary components of thecomplex number z in the form:

cos$ =x

#hence x = # cos $

sin$ =y

#hence y = # sin $


As z = x+ iy in Cartesian notation, we can then express z in the form:

z = #(cos$ + i sin $)

Moreover, Pythagoras’ relation for the right triangle indicates that:

cos2 $ + sin2 $ = 1

which may be decomposed into:

(cos$ + i sin $)(cos$ # i sin $) = 1

Let us thus find the expression for a product of the type: (cos$ + i sin $)(cos! +i sin !).

We have:

(cos$ + i sin $)(cos! # i sin !) = cos $ cos! # sin $ sin!

+i(cos $ sin! # sin $ cos! )

However, based on the well-known trigonometric formulas:

cos $ cos! # sin $ sin! = cos($ + !)

cos $ sin! # sin $ cos! = sin($ + !)

We thus obtain:

(cos$ + i sin $)(cos! # i sin !) = cos($ + !) + isin($ + !)

Let us then posit that $ = !. It follows:

(cos$ + i sin $)2 = cos(2$) + isin(2$)

We can thus easily conclude upon the formula proposed by Abraham de Moivre(1667–1733) and rediscovered by Euler in 1748, in his famous text Introduction to theAnalysis of the Infinite:

(cos$ + i sin $)n = cos(n$) + i sin(n$)

Around 1730, de Moivre also found the inverse formula:

n!(cos$ + i sin $) = cos($ +

2k"

n) + i sin($ +

2k"

n)


This formula makes it possible to find trigonometric solutions to equations of thetype:

Xn # 1 = 0

These solutions, the nth root of unity, which is identical to the quantity cos($ +2k!n )+ i sin($+ 2k!

n ), are distributed equidistant from one other on the trigonometriccircle.

6.4. Consequences on Hegelian philosophy

This Gaussian advance would allow Hegel to get past the extremely negativeopinions that he had first held on mathematics in Phenomenology of the Spirit and inhis pedagogic texts. One text in particular, taken from The Science of Logic, entitled“The Doctrine of Concept”, perfectly explains this volte-face.

This text relates to the idea of knowing and is found in the second chapter of thethird section of the Doctrine of Concept, which talks about Subjectivity, Objectivityand Idea. The concept of ldea splits itself into three subconcepts: Life, the Idea ofKnowing and the Absolute Idea. The Idea of Knowing is thus determined by the Ideaof Truth and the Idea of Good. And the idea of Truth is differentiated into Analyticalknowing and Synthetic knowing, which contains within itself definition, division andtheorem.

The concerned text is located at the end of Analytical Knowledge and is atransition toward Synthetic knowledge. His central idea is that the use of complexnumbers and their trigonometric representation by Gauss in order to resolve certainalgebraic equations perfectly illustrates the possibility of associating a problem withsolutions that are not contained in the given data for the problem. As Hegelcomments, this type of procedure would incline one to contest that arithmetic was asolely analytic science. This is one of the points where the traditional criticism thatthe philosopher raises against mathematics stumbles. Here is another point whereanalysis overtakes him toward synthetic thought that appears, at its limit, to identifycontradictions (line and curve):

“In the higher analysis, where with the relationship of powers, we are dealingespecially with relationships of discrete magnitude that are qualitative and dependenton Notion determinatenesses, the problems and theorems do of course containsynthetic expressions; there other expressions and relationships must be taken asintermediate terms besides those immediately specified by the problem or theorem.And, we may add, even these auxiliary terms must be of a kind to be grounded in theconsideration and development of some side of the problem or theorem; the syntheticappearance comes solely from the fact that the problem or theorem does not itself


already name this side. The problem, for example, of finding the sum of the powersof the roots of an equation is solved by the examination and subsequent connectionof the functions which the coefficients of the equation are of the roots. Thedetermination employed in the solution, namely, the functions of the coefficients andtheir connection, is not already expressed in the problem - for the rest, thedevelopment itself is wholly analytical. The same is true of the solution of theequation x(m&1) # 1 = 0 with the help of the sine, and also of the immanentalgebraic solution, discovered, as is well-known, by Gauss, which takes intoconsideration the residuum of x(m&1) # 1 divided by m, and the so-called primitiveroots – one of the most important extensions of analysis in modern times. Thesesolutions are synthetic because the terms employed to help, the sine or theconsideration of the residua, are not terms of the problem itself” [HEG 16, pp.265–266].

This remarkable text reveals a Hegel who is confronted with mathematics that, inhis most advanced problems, goes beyond the conditions that initially characterized itas a purely analytical domain and one in which understanding, based on externalreflection, proceeded to catastrophically dismember the real, which resulted inleaving things completely disparate instead of presenting a veritable logicalexplanation for their concrete development. This was, for Hegel, the usual procedurein geometry, notably in the steps for a proof. This was seen, for instance, with respectto the demonstration of Pythagoras’ triangle, a famous case discussed in the prefaceto Phenomenology of the Spirit.

In the case of Pythagoras’ triangle, the proof that the square of the hypotenusewas the sum of the square of the other two sides of a right triangle was obtainedby constructing real squares on the sides. The areas of these squares would then becompared to that of the square constructed on the hypotenuse. According to Hegel, thisprocedure was purely analytical and completely dismembered the object, the “righttriangle”, which, for the needs of this cause, was broken up into three squares that hadvery little to do with it.

On the contrary, in the case of an equation of the type xn # 1 = 0, the solutions,even if purely analytical, do not follow from the nature of the problem itself.Algebraically speaking, the polynomial must be decomposed over C, which can beobtained by factoring x # 1 and by expanding the n # 1 solutions that remain in theform of n # 1 complex numbers. This results in Alembert’s theorem, whichstipulates that a polynomial is always decomposable over C where its correspondingequation accepts n distinct solutions.

However, Gauss–Argand’s true invention consisted of adjoining a trigonometricrepresentation (the Gauss–Argand plane) to complex numbers. This made it possibleto situate them on a circle – the trigonometric circle – as the points with coordinates


cos( 2k!n ) and sin( 2k!n ). In the case of the equation xn # 1 = 0, the solutions are ofthe form:

x = cos(2k"

n) + i sin(

2k"

n)

The detailed analysis of Gauss’ procedure1 in Recherches arithmétiques(Arithmetic Research) [GAU 89, pp. 433–479] yielded the result that, as themathematician stated, “the solution of the equation xn # 1 = 0 may be reduced tothe solution of the n # 1 degree equation with two terms xn&1 # T = 0, where T isdetermined by the roots of the equation xn&1 # 1 = 0” [GAU 89, p. 478]. Hegel onlychanged the m to n and his expression, which was concise at best and wrong, atworst, if we take it literally, thus bring us back to this method of Gauss’ which heknew of and in which he recognized the advance that it represented for arithmetic,even the resulting closeness to philosophy: mathematics that had, until then, beenpurely analytical now grew more synthetic and thus became analytico-synthetic, acharacteristic that was, until this point, only associated with purely conceptualthought.

6.5. Euler’s formula

In a purely mathematical context, the fundamental formula that made it possibleto associate five apparently unrelated numbers had been proposed many years earlierby Euler, universal and synthetic mathematician par excellence. These five numberswere: e, i,", 0 and 1. Knowing that:

ei" = cos$ + i sin $

when we replace $ by ", it follows:

ei! = cos" + i "

But:

cos" = #1 and sin" = 0

Hence:

ei! = #1 [6.1]

1 See our comments in [PAR 93c].


or again:

ei! + 1 = 0

6.6. Euler, Diderot and the existence of God

In his memoirs, French gentleman Dieudonn Thiébault (1733–1807), recounts ameeting that took place in Saint Petersburg in 1770 between the French philosopherDenis Diderot and a certain mathematician who seems to have ridiculed him.

Diderot, who was there on the invitation of the Russian empress Catherine II(“Catherine the Great”), had been received very warmly and “fully provided for bythe Empress, who greatly enjoyed the fertility and warmth of his imagination, theabundance and uniqueness of his ideas, and by the zeal, boldness and eloquence withwhich he publicly preached atheism”.

At the instigation of some old courtiers, who were alarmed by the possibleconsequences this doctrine might have on the youth, it was decided they must forceDiderot into silence. The empress did not wish to be the source of this censure andresorted to a subterfuge. Diderot was informed that a Russian philosopher,“mathematical thinker and a distinguished member of the Academy” had offered toalgebraically demonstrate the existence of God before the court.

The philosopher agreed to hear him out and, when the time came, thepseudo-Russian (none other than Euler, Christian mathematician! His presence inSaint Petersburg at this period has also been attested) approached him and, in a toneof conviction, challenged him thus, “Monsieur, (a + bn)/z = x; therefore Godexists. Respond”. Not knowing what to say, and fearing other sessions of a similarnature, Diderot is thought to have rapidly communicated his desire to return toFrance, a wish that was speedily granted.

This anecdote, reported by Thiébault [THI 04, p. 141], was then repeated, withsome variations, by many different authors – DeMorgan, Cajori, Bell and Hogben,among others – before Gillings [GIL 54] really reviewed the text, this time correctingthe formula that had, all this while, been wrongly reported (the z of the denominatorhad been replaced by n).

The reinstatement of the z into the equation is, perhaps, worthy of comment, asuntil then the mathematical expression had been assumed to have no meaning. Let usthus start from the beginning. We have:

a+ bn

z= x


The choice of z as the denominator seems to suggest that x (= God?) is in fact theratio of a real number to an imaginary number. We can, in effect, write:

z = a+ ib = #(cos$ + i sin $) = #ei" hence: a = # cos $, b = # sin $

It follows:

x =# cos $ + #nsinn$

#ei"

which is another way of writing:

x =a+ bn

a+ ib, [6.2]

and which we can further reduce, for # $= 0, to:

x =cos$ + #n&1sinn$

ei"[6.3]

We then observe that for $ = 0 or $ = ", Euler’s formula makes it possible todeduce x = 1 (God is One?), which also corresponds to a = 1, b = 0 in Formula[6.2], while the hypothesis a = 0, b = 1 would result in:

x =1

i

which corresponds to $ = "/2 and #n&1 = 1 in formula [6.3].

But it is evident, assuming the anecdote is true, that this was really more of a jokethan “proof”.

6.7. The approximation of functions

The demonstration of formula [6.1] assumes the concept of “limited development”of a function, which was introduced by Taylor and Mac Laurin’s work in the 18thCentury. They were the first to find an approximate expression for functions in thevicinity of certain points.


6.7.1. Taylor’s formula

Knowing the derivative of a function f(x), f !(x) is expressed in the followingmanner:

f !(x) = limx"x0

f(x)# f(x0)

x# x0

We demonstrate that it is possible to posit:

f(x) = f(x0) + (x# x0)f!(x0) +

(x# x0)2f”(x0)

2!+ ...

+(x# x0)nfn(x0)

n!+O(xn)

fn(x0) is the nth derivative with respect to the point x0.

O(n) is a function that tends to zero when x tends to zero.

6.7.2. MacLaurin’s formula

By positing x0 = 0, MacLaurin observes that the formula can be simplified to:

f(x) = f(0) + f !(0)x+f !!(0)x2

2!+ ...+

fn(0)xn

n!

This formula is all the more useful given that the derivatives at the point x = 0often have simple values. Thus, for a trigonometric function, we can easilydemonstrate that we have, notably:

cos x = 1# x2

2! +x4

4! # . . .

sin x = x# x3

3! +x5

5! # . . .

Let us rapidly demonstrate this for cos x. We know that:

(cos x)! = #sin x

thus:

(cos x)!! = #cos x, (cos x)!!!

= sin x, etc.


For x = 0, we thus have, successively:

cos x = cos(0)# sin(0)x# cos(0)x2

2!+

sin(0)x3

3!+

cos(0)x4

4!# etc.

= 1# x2

2!+

x4

4!# . . .

On multiplying sin x by i, we have, correlatively:

i sin x = ix# ix3

3!+

ix5

5. . .+ . . .

Hence:

cos x+ i sinx = 1 + ix# x2

2!# ix3

3!+

x4

4!+

ix5

5!# ...

We also have, for the exponential:

ex = 1 + x+x2

2!+ . . .+

xn

n!

Hence:

eix = 1 + ix+i2x2

2!+

i3x3

3!+

i4x4

4!+

i5x5

5!# ... = 1 + ix# x2

2!# ix3

3!

+x4

4!+

ix5

5!# . . .

and finally:

eix = cos x+ i sin x

The remarkable connection that Euler established between these fivemathematical constants, e, i,", 0 and 1, necessitates a deep philosophical meditation.Mathematics establishes relations between entities that were believed a priori to beindependent, and this thus makes it possible to introduce to the world a unificationthat was unperceived earlier. In this sense, it seems to serve the synthetic perspective,which, at least on the continent, is that of philosophy and thus, philosophy cannot inany way give this up or consider it to be negligible.


6.8. Wronski’s philosophy and mathematics

While the unifying aspect of Euler’s formula has, seemingly, escapedphilosophers, Taylor and MacLaurin’s formulas (which, as we have seen, made itpossible to construct the “limited development” of functions in the vicinity of certainpoints) had several philosophical consequences.

Joseph Marie Hoëné-Wronski (1776–1853), a philosopher and mathematicianborn in Poznan (Poland), had abandoned a military career in 1803. Following a7 year stint at the Marseille Observatory, where he pursued his mathematical andphilosophical reflection, in 1810 he addressed a dissertation to the Institut de Francetitled: Principes premiers des méthodes algorithmiques (The first principles ofalgorithmic methods). This was favorably received by Lagrange. This works allowedhim to develop a completely original conception of the Absolute: the Absolute as auniversal law of development2.

According to Wronski, the need for unity of the Absolute as well as that of theuniverse implies that the generation of different systems, of beings or connections ofknowledge, follows one and the same rule (or the law of creation)3. If we knew ofthis law of creation, on which everything depends, we can then deduce from this, apriori, the general construction of different systems of being or facts of knowledgethat make up the universe and constitute the object of diverse sciences andphilosophies. We can thus obtain, based on this, the general construction of diversescientific and philosophical systems and, thus, the general architectonics of allsciences and of philosophy. The law of creation is, therefore, just as much a law ofcreation of beings as a law of creations of knowledge objects [HOE 75, p. 19].

Given the fundamental identity of the being and knowledge that defines theAbsolute, this law of creation is the law of creation of the Absolute by itself. TheAbsolute is self-produced (autogeny) in and by itself, and at the same timeestablishes itself (authothesy):

2 Let us recall here that philosophers had only four conceptions of the Absolute: the Absoluteas cumulative totality (from the Sophists to the Encyclopedists); the Absolute as a stable subset,eventually reduced to a single invariant, whether situated external to the world or in the world(Good, God, ideas, universal and fundamental categories, substance, etc.); the Absolute asa period or cyclic process that leads to the recurrence of the same series (Hegelian concept,the Pythagorean theory of the eternal return etc.); and finally, the Absolute as pure difference(contemporary philosophies).3 Wronski’s work was annotated from the early 20th Century onwards (see [WAR 25, ARC70b]). The mathematical aspect of some of his discoveries resonate even today (see, forexample, [LAS 90, pp. 379–386]).


“It would thus be enough to operate the genetic development of the Absolute itselfto obtain knowledge of this important law of creation, the possession of which, whenjoined to the knowledge of the Absolute, gives man, we dare to say, the all-powerfaculty of assisting God in creation, of reproducing the creation of the universe and,finally, carrying out his own creation” [HOE 81, p. 75].

But contrary to what the word “Messianism” may suggest, there is nothingmystical about Wronski. “When you understand the reason of the universe, you canspeak of God, faith and religion”, he states. “Do not do this before” [HOE 79,p. 239].

6.8.1. The Supreme Law of Mathematics

We will see that Wronski used the expression “genetic development of theAbsolute”. Such a formula could never be limited to a metaphor for this philosopher,who believed that reason, guided by mathematics, commanded the world. In reality,in the period that Wronski wrote in was a period that saw a lot of work in analysis inorder to develop functions into series [MCC 81, pp. 16–24]. The law of creationcould only evoke the particular series that Wronski authored and which he called,slightly pompously, “The Supreme Law of Mathematics”. This was expressed in thevery general form given below (we have replaced the upper case letters with lowercase letters and write the functions in accordance with present norms):

f(x) = a1(1(x) + a2(2(x) + ...+ an(n(x) + ...+ etc., [6.4]

where f(x), (1(x), (2(x), ... are arbitrary functions of x, the coefficients ai beingindependent of this variable.

A serious study of Wronskian mathematics makes it possible to go beyond thesometimes condescending commentaries penned by some mathematicians around hisformula4.

It is, in effect, Wronski’s most notable mathematical contribution and, while it isindeed a formula for the coefficients of development of a function f into a series offunctions (1(x),(2(x), ...,(n(x), its proof consists of taking systems of equationsformed by the function f and its derivatives of any order, while implicitly assumingthat these derivations are possible.

4 Philip J. Davis and Reuben Hersh suggest that it may, thus, be a sort of generalized Tayloriandevelopment that contains all past and future developments [DAV 82, p. 58].


To derive the formulas for the coefficients ai, Wronski used, in particular,combinatorial sums in the form of determinants. This later led to Thomas Muir [MUI82, p. 224] calling such determinants “Wronskian”5, a term that is still used today.

We know that Wronski’s “Supreme Law” had drawn Cayley’s attention6, whoproved a certain version of this, and then that of Charles Lagrange [LAG 96], whoprovided a conventional treatment with an error term and convergence conditions.Finally, Stefan Banach [BAN 39] himself presented a modern functional analyticversion7.

In the years following its formulation, Wronski’s law gave rise to different,particularly in-depth studies [WES 81, WES 82, ECH 93], which today make itpossible to clarify its meaning.

Let us also note that from a current-day point of view, there are times when hisequation is greatly simplified. For example, if the series of functions (1(x), (2(x), ...forms an orthonormal base with respect to the standard product, or, indeed, any interiorproduct (., .) in the vector space of infinite dimension of single-variable polynomials,then for each i we have:

ai = f(x),(i(x)

This kind of a situations is, however, rare. Thus, what Wronski called “theSupreme Law of Mathematics” is none other than the method that makes it possibleto calculate the coefficients ai in the general case. While developing this method heused, as auxiliary objects, specific determinants that have since been called“Wronskians” and we will see that they underpin this law.

A “Wronskian” is the determinant of a matrix constructed by placing thefunctions in the first row, the first derivative of each function in the second row andso on through the (n # 1)th derivative and thus forming a square matrix, sometimescalled a “fundamental matrix”. This assumes, of course, that the functions in question

5 This seems to be the first usage of the word (see [CAJ 19, p. 340]).6 A. Cayley [CAY 73] repeated in [CAY 96, pp. 96–102].7 We know, today, that the convergence problems that Wronski’s formula brings up can beresolved if we operate precisely within what has since been called “Banach’s space” (see[PRA 08]).


are (n # 1) times differentiable. In modern notation, the Wronskian of n functions(1(x), (2(x), ...,( n(x) can be expressed as follows:

W ((1,(2, ...,(n) =

***********

(1 (2 · · · (n

(!1 (!

2 · · · (!n

(1” (2” · · · (n”...

.... . .

...(n&11 (n&1

2 · · · (n&1n

***********

Today, Wronskians are commonly used in different parts of mathematics (analysis,invariant theory, algebraic theory of binary forms, algebraic geometry, etc.) which isproof that Wronski managed to alight upon a particularly significant object. They aremost often used as a test for linear independence, a procedure related to the followingresult.

THEOREM.– Let us assume that (1, · · · ,(n are (n#1)-times differentiable functions.If W ((1, · · · ,(n) is not identically null, then the functions (1, · · · ,(n are linearlyindependent.

The problem is different in the case of Wronski’s supreme law, as Echolsdemonstrates. Assuming that the series is infinite and true, Wronski tried todetermine the form of the constant coefficients ai. This is what he did over the last 90pages of his Philosophie de la technie algorithmique (Philosophy of AlgorithmicTechnique) His method, which was long and intricate, can fortunately be shortenedthanks to the modern use of determinants. According to Echols, the generalcoefficient an takes the form:

an =#n(f(a))

#n((n)#

µ=x$

µ=n+1

aµ#n((µ)

#n((n)[6.5]

where the variable x has been replaced by an arbitrary value a, with:

#(F ) =

*******

(!1 · · · (!

n&1, F !

......

. . ....

(n1 · · · (n

n&1, Fn

*******

the symbol (r signifying that after having differentiated ( r times, x was changed toa. We will see, however, that Wronski did not stop with the operation ofdifferentiation using, in fact, a symbol for the repetitive operation of which derivationand differentiation are only particular instances.


Be that as it may, the value of each coefficient in series [6.4] is given in [6.5] interms of an infinite series containing all the coefficients that follow. As the series isinfinite, it is non-exhaustible. Through successive substitutions, an can be expressedas a series whose terms contain coefficients of as high an order as we desire. Given thenature of this expression, it may seem that the value of an must remain indeterminate,as long as there are no means to test its arithmetic equivalence. But we also see that ifwe give µ a finite limit value m, it is easy to see that [6.5] becomes:

an = (#1)n

******

(!1 · · · (!

n&1 (!n+1 · · · (!

m&1 f !(a)· · · · · · · · · · · · · · · · · · · · ·(m&11 · · · (m&1

n&1 (m&1n+1 · · · (m&1

m&1 fm&1(a)

************

(!1 · · · (!

m&1

· · · · · · · · ·(m&11 · · · (m&1

m&1

******

[6.6]

As Echols shows again, the only reason that could explain why Wronski did notgive this general form to his coefficient is that at the time he was writing in, he didnot know how to evaluate this relation when m became infinite8. In the case of theWronski problem, however, the question can be easily resolved and it is almost strangethat this was not done.

In reality, if the functions ( of equation [6.4] are such that (rq disappears for r < q,

either by operations carried out on the functions, or by substituting particular valuesfor the variable, then the last column of #n((µ), (µ = n+ 1, ...,*) also disappears,like the second term on the right-hand side of [6.3], such that an is reduced to:

an = (#1)n

********

(!1 0 · · · 0, f !

(1” (2” · · · 0, f”· · · · · · · · · · · · . . .(n1 (n

2 · · · (nn&1, f

n

****************

(!1, 0 · · · 0

(1”, (2” · · · 0· · · · · · · · · · · ·(n1 , (n

2 · · · (nn

********

[6.7]

which is the form of the coefficients in [6.4]. If we then posit (n = (!(x))n andmake !(a) = 0, we deduce the Wronski formula for the Burmann series, which itself

8 A Jacobi theorem has since simplified this writing, making it possible, under certainconditions, to remove the minors of a functional determinant.


includes the Lagrange series and the Laplace series and, consequently, also the Taylorseries.

6.8.2. Philosophical interpretation

Wronski’s supreme law is, in fact, a linear combination of functions that are thelocal solutions of a differential equation of the nth order. Inasmuch as it exists, thisequation corresponds to a non-null determinant, the “wronskian” of these n functions.The question that remains is: why does this formula seem to symbolize the law ofcreation and the very form of the Absolute?

We must, in fact, imagine that for Wronski, the world is a flow whose specificdevelopments we are aware of each time. We perceive a set of forms and diversemovements. The law of production of this flux is thus given as the series of particularsolutions whose combination may be considered as the solution of a generalizeddifferential equation.

The existence of this series and its particular solutions, which expresses thecondition of possibility of a linear system of n functions, is but the translation of thenon-annulation of the Wronski determinant (or wronskian). As we have seen in6.8.1., if the Wronskian is not null, the n functions of the linear system, whosecoefficients have been calculated by Wronski, are independant and form the base ofan immense functional space, the vector space in which reality may be decomposed.

In other words, wherever classical philosophy defines the Absolute either as a pointof accumulation or as a stable subset or, again, as the recurrence of the same temporallogic, Wronski operates a sort of functional decomposition of reality. He brings thisto a combination of elementary functions with a non-null infinite determinant as thecondition for existence.

This condition, quite evidently, bears no comparison to the naive representationsof the Absolute in classical philosophy (including Schelling and Hegel’srepresentations). Knowledge, Wronski believed, must be expanded to infinity, withthe development of the series in the supreme law:

“The general object of philosophy is to deploy, across the infinity of its expanse thisspontaneous and limitless activity of man’s Knowledge, to reveal the entire creationof the universe, from the very creation of God to the individual creation of man [...]The general purpose of religion is to transform the precarious and ephemeral beingof man, dependent on physical conditions and perishable, into an Absolute being, inaccordance with their absolute Knowledge and independent of any condition foreignto their own, infinite absolute Knowledge” [HOE 42, pp. 178–179].


The independence of knowledge with respect to the being is translated by thepolicy of absolute antagonism (or social antinomy) in two parts: that of thesovereignty of the people and divine sovereignty. Wronski laments that these areirreconcilable.

However, leaving aside the political aspects of this thinking, the major conclusionwe note is that for Wronski, the fundamental law of history (a truly functionalcombination of particular solutions whose existence we observe at each instance) isdefinitely based on the existence of a non-null infinite determinant, formed by aninfinite set of elementary functions and their infinite derivatives. A sort of “controlpanel” for God.

We see how and to what extent such a combination is abstract and how it alreadycontains much more formal and conceptual richness than Hegelian “logic” and itsternary sequencing. This holds, moreover, regardless of the number of taxonomicallevels in play.

This conception makes it possible to imagine, in a non-trivial manner, the conceptof independence based on the model of linear independence regardless of thegeneralized mode. Wronski’s philosophy is able to simultaneously conceive of thedependence and independence of man with respect to God and the fact that Godcreated all creatures.

6.9. Historical positivism and spiritual metaphysics

6.9.1. Comte’s vision of mathematics

As we know, in the classification of sciences that he proposes and that underpinsthe organization of lessons 1 to 45 of his Course of Positive Philosophy, Comte [COM75] accords mathematics’ prime position.

What does he mean here by “mathematics”? Comte distinguishes, within thisdepartment, an abstract part (formed of purely computational tools) and a concretepart (made up of its different domains of applications: geometry, mechanics andastronomy). We will be concentrating, here, on the first part, setting aside the secondpart (although Brunschvicg did think that the second part formed “the center ofgravity” of this work).

“Admirable extension of the natural logic to a certain order of deduction”[COM 75, 2nd lesson, p. 64]. For Comte, mathematics in its abstract part, that iscalculation, is not only the component part of natural philosophy but the base ofpositive philosophy. The starting point, in truth, for any general or specializedscientific education.


The universal use of this discipline, he says, has been seen over the course of timeand thus justifies this primordial position accorded to it. But we can go further. “Intruth”, writes Comte, “it is through mathematics that positive philosophy began to beformed: it is from mathematics that we acquired the method” [COM 75, 3rd lesson,p. 82].

These abstract mathematics, described in lessons 4–9, are essentially arithmetic(or the computation of values) and algebra (or the calculation of functions). But as thefirst calculation, with the development of mathematical science, was no more than anappendix to the second, it eventually disappeared as a distinct section into the body ofabstract mathematics [COM 75, 3rd lesson, p. 91]. From this situation, it follows thatthe method of positivism must be entirely contained in the calculation of functions.

The calculation of functions is composed of two fundamental branches, Comteexplains. One aims to resolve equations established between quantities themselvesand the second, “starting from the equations, much easier to form in general, betweenquantities that are indirectly related to those in the problem, this constant objective ofthis branch is to deduce, corresponding equations between the direct quantities thatwe consider using invariable analytical procedures”. This brings the second questionin the same domain as the first [COM 75, 4th lesson, p. 95]. The first calculation isordinary analysis (or algebra), the second is transcendental analysis (in other words,the infinitesimal calculus). Comte, aiming to generalize and clarify Lagrange’s ideas,would rechristen these: the calculation of direct functions and the calculation ofindirect functions.

The calculation of direct functions can be further divided into two parts: thealgebraic resolution of equations, satisfactory in itself, and the numerical resolutionof equations, an incomplete and bastardized operation that mixes algebraic andarithmetic questions, but presents obvious practical advantages. These two parts aredominated by a third, purely speculative part, called “theory of equations”, whichrelates to both the composition of equations as well as their transformation.

Comte concluded his study of the calculation of direct functions with someobservations on imaginary numbers, negative quantities, the relations between theconcrete and the abstract and the theory of homogeneity (though it is difficult tounderstand what we can draw from this from a methodological point of view).

The philosopher then dedicates two lessons (the 6th and the 7th lesson) to thecalculation of indirect functions. They begin by describing the three historic formsthis has taken (Leibniz, Newton, Lagrange) before going on to discuss each part indetail (differential and integral calculus).

In the following studies on the calculation of the maxima and minima of afunction (calculation of variations) and on the Taylor methods (calculations using the


finite difference method), it is still difficult to see a philosophy emerging. The onlypossibility is that this philosophy is restricted to retranscribing, using largercategories of reorganization, which Lagrange has already stated so well in hisAnalytical Mechanics.

These same remarks can be made with respect to most of the lessons concerningconcrete mathematics. We must, however, make an exception for the end of the 14thchapter, which discusses the classification of Monge surfaces and which has a generalcharacter that Comte refers to endlessly later on.

The problem in classifying surfaces resides in the incompatibility of their naturaldivisions based on their mode of generation (cylinders, conics, of revolution, etc.)and their analytical expression, with classification based on the degree of theequations destroying the former (for instance cylindrical surfaces may be of allimaginable degrees). Monge resolved this question by observing that, “surfaces thatare subject to the same mode of generation or necessarily characterized by a certaincommon property of their tangent plane at some point” [COM 75, 14th lesson,p. 223]. Thus, analytically expressing this property based on the general equation ofthe plane tangent to some surface, we form a differential equation that represents allthe surfaces of the same family.

This search for an invariant is precisely what would be demanded of naturalists inzoology and botany. Comte continues to refer to Monge in the 40th lesson, dedicatedto general considerations on biological sciences, as if this was inspired by the “safezoological and botanical methods”.

Comte’s attitude is, therefore, ambiguous. On the one hand, his paradigm here isbiological. On the other hand, mathematics emerges with a degree of additional puritythat can be understood elsewhere. This must, thus, be studied for its simplicity andbecause it presents the model with a certain clarity.

Drawing out real analogies between phenomena is, moreover, a specific propertyof mathematical analysis. Nowhere is this more clearly demonstrated than inFourier’s [FOU 88] theory on heat, about which Comte, in the 31st lesson, isabsolutely rhapsodic. He claims, notably, that “no other mathematical creation hashad greater value and significance than this as regards the general progress of naturalphilosophy”. [COM 75, 31st lesson, p. 513]. Bachelard [BAC 28] would laterenhance this further by noting this important fact: the initial conditions for thepropagation of heat assume arbitrary functions, whose forms are, nonetheless,sinusoidal. It remained to be explained that under given conditions, any sufficiently


limited function could take sinusoidal form9. This algorithm, that we now call “theFourier transform” and which is expressed as the “infinite sum” of trigonometricfunctions of all frequencies, has since had a remarkable future, eventually becomingan essential tool of harmonic analysis. Indeed, in this discipline, the “Fouriertransform” is an extension, for non-periodic functions, of the series development ofFourier’s period functions. The Fourier transformation associated with an integrablefunction is defined over the set of real numbers or that of complex numbers, afunction called “Fourier transform” whose independent variable may be interpretedphysically as the frequency of pulsation, but also as a “specter” when the state of thefield at a point is represented by a “signal” function.

Could these methods be used in philosophy? In mathematics (as in botany or inzoology) analogies must be identified between structures that make it possible toconstruct a valid classification. As L. Brunschvicg observed, although Comte’sphilosophy does not open onto metaphysics, the last lesson in the Course on PositivePhilosophy ends with the following idea: instead of searching for a sterile scientificunit by reducing the phenomena to a single order of laws, it would be better toexamine the various classes of events as being equipped with special laws. However,having said this, these laws are “inevitably convergent and, in some respects,analogous”. The Course thus also leads to a categorization of classes of events andtheir respective knowledge, and is organized into an immense encyclopedia. But, ofcourse, as M. Serres demonstrates, this did not anticipate the future and, instead,described a science that has since died out: that of the 18th Century.

6.9.2. Renouvier’s reaction

Contrary to Comte, Charles Renouvier (1815–1903), expressed the strongestreservations toward the idea of taking mathematics as the model for thought. Hechallenged even the plausibility of applying mathematics to the world and to nature.

Thus, he wrote La Nouvelle Monadologie (The New Monadology), a book inwhich he rectified certain Leibnizian theses, that laws are indeed, “in concrete order,resulting from general relations of causality established between phenomena thathave their seat in monads”. However, “there is nothing that binds these ratios toobserve rigorous, mathematical exactitude, commanded by the abstract, mother ofthe sciences” [REN 99, p. 50].

A note further clarifies Renouvier’s point of view: “If we compare the concept wemay have of an insect or a bird, artificial automata with the most marvelous

9 This manner of reconciling the non-periodic with the periodic would find in Bachelard anattentive reader who was able to use such a model in philosophy (see [PAR 07]).


construction imaginable, with the experience of the real animal and its movements,we would indeed find it difficult to understand that something that is similar towhatever is in the first is also in the second and, taken to the Absolute, that is withevery flap of the wings or its degree, calculable in itself to the last degree ofprecision, or is given in the cause-effect relation. What will become of this if everycause must be related with the same mathematical rigor to all the earlier causes andcoefficients, given in the living molecules and in those things which must also bemade mathematical functions? It has always appeared natural to envisagespontaneous movements that begin from somewhere and stop somewhere, sensibly,and do not suffer the application of an exact measure. Chaining these mathematicallyto causal powers in the universe will enable neither beginning nor end to anythingand would, at the same time, enjoin strict measurement for each phenomena, as theabstraction would wish it, introducing inflexibility in change and would be a negationof life” [REN 99, pp. 87–88].

As L. Fedi, shows, “While Comte searched for an archetype of positive inmathematics, Renouvier recommended modern physics and chemistry” [FED 01].He also remained so guarded in these references that while praising the contributionof science, and adopting the Reason that accompanied them, he would also restrictits reading such that despite his excellent education (he was a graduate of theEcole Polytechnique) he bypassed precisely those extensions of rationality thatcharacterized it. Thus he refuted not only the concept of actual infinity, accepted byLeibniz and soon developed by Cantor, but also challenged the pertinence ofnon-Euclidean geometry which, according to him, strayed from the real stricto sensu.

Even while attempting to share his mistrust of a certain mathematical mysticism,he ended up refuting that which he called the “confusion” of natural mathematics, “anidolon specus that arises among physicists from the habit of mathematical methodsand, among philosophers, from the misapplication of these methods” [REN 99, p.89]. Science, as he saw it, and from which he excluded all that was accidental anddisturbed, could not coincide with nature of which science was, in reality, only anabstraction.

Renouvier’s philosophy thus modified Leibniz in the wrong way. Where Leibnizadmitted infinitesimals in actuality and conceived of disorder as a simpledeformation of order, thus anticipating both topology as well as non-standardmathematics, Renouvier stated a blanket rejection of all these overtures and restrictedhimself to an overly pragmatic view of knowledge.

6.9.3. Spiritualist derivatives

French mathematical philosophers of the 19th Century do not seem to have beenvery fortunate in their judgments. Battling too hard to defend spiritualism (something


that Ampere never stopped doing) or too concerned with reconciling science andreligion (which soon became Boutroux’s project), they often had too narrow a visionof their discipline, like Milhaud, unable to see any part of what would form the futureof this field. Thus, they remained positivist even when they believed they had turnedtheir back on Comte. Studied independently of its future or as a catalogue of result,the sciences serve no philosophical purpose. They are only useful to philosophywhen they are clarified by epistemology.

When this was not the case and when, on the contrary, philosophy becametheology, or lost itself in unrestrained speculation, there clearly remained nothing ofsubstance. Thus, the series of obscure ex-students of the Ecole Normale Supérieurethat Ravaisson described in La philosophie en France au XIXe siècle (Philosophy inFrance in the 19th Century) [RAV 84], did not really make a stir in the world. At thesame time that science was refining and complexifying itself, French philosophy,unable to distinguish the major axes of knowledge, took refuge in the mosthackneyed idealism and spiritualism, splitting into a plethora of little works,generally of very limited interest, while also sometimes illuminated through the useof ineffective pseudo-mathematics (P. Gratry intended to use infinitesimal calculus toreach God! [RAV 84, pp. 142–145]).

6.10. The physical interest of complex numbers

Let us now return to complex numbers. What could they be used for? Given thatthey are imaginary, should we not think that they have no relation to the real? Aconclusion of this kind would be a grave error. Complex numbers have, for a verylong time now, been indispensable to the physicist.

The trigonometric interpretation of complex numbers must be used here as a guide.By positing z = #(cos$ + i sin $), we have seen that we can characterize a point Min a plane XOY, based on the modulus #, which expresses the distance of M from theorigin and the angle $, which intervenes as an argument. This is, however, still a staticrepresentation. We can develop a more ’dynamic’ interpretation of complex numbersby observing that the modulus # can be used to define a position on a trajectory andthat the argument $ may then indicate a measure of the rotation which affected aninitial trajectory.

It was indeed one such interpretation that led the physicist Augustin Fresnel(1788–1827) to the first use of complex numbers in his theory on optical reflection.Fresnel, according to Huyghens, was at the origin of a wave-representation of light,for which he presented the first coherent mathematization. Fresnel postulated thatlight was composed of vibrations that were transverse to its propagation, which hecompared to the elastic transverse vibration of solids. In reality, this kind of purelymechanical theory would later be replaced by Maxwell’s electromagnetic theory,


where light is due to the simultaneous propagation of an electric field and a magneticfield, the vibrations of the electric field representing the luminous vibration in thespace where light is propagated. A luminous vibration at a point in space isrepresented by a vector whose point of origin is this point. The extremity of thisvector thus describes a certain curve in a plane perpendicular to the direction ofpropagation. The projection along an axis of this plane then gives a waverepresentation as a periodic function of time.

As the simplest periodic function is the sinusoidal function, we represent theluminous vibration as a sinusoidal function of time (see Figure 6.2).

Figure 6.2. The luminous vibration

The equation of the wave is written, starting from the origin, as

x0 = a cos2"

)= a cos 2" Nt = a cos ( t

– a is the amplitude of the vibration;

– 2"Nt = 2!# t is the pulsation;

– N is the frequency;

– ) is the period.

We have N = 1# (the frequency is equal to the inverse of the period).


A complex representation of the wave thus takes the form:

x0 = a (cos (t + i sin ( t) = a exp(i(t)

This is justified above all by the fact that when using De Moivre’s and Euler’sformulae, the multiplication and exponential calculations of such formulas are foundto be greatly simplified.

For the same reason, the mechanical quantity, starting from Schrödinger,commonly used complex waveform representation.

Generally speaking, the use of complex numbers has been justified in severalstudies of hydrodynamics, where rotations are frequently produced (whirlpools,vortices) as well as the establishment of dynamic models for the formation ofstructures that possess symmetries of rotation, including the local plane.

6.11. Consequences on Bergsonian philosophy

One of the philosophical consequences is as follows: as Claude–Paul Bruterobserves, the term “hydrodynamic” must not be limited solely to the context of usualfluids (water, oil, wine, etc.). As Wronski implicitly suggests, the world can beconceived of as a sort of stratified fluid whose local movement sometimes freezeswith greater or lesser slowness onto particular attractors, forming frames ofmomentarily rigid objects at our scale of observation in the universe.

Maxwell’s electromagnetism gave rise to representations of this type, which werethen generalized. Toward the end of the 19th Century, under the influence ofW. Thomson (Lord Kelvin) and physicists such as W. Ostwald, this waveform andfluid representation of the universe was transposed onto the entirety of reality withthe name “energetism”. The atom, as a material point, was integrated into a theorycalled “atom-vortex” theory, which viewed this as a small, fluid whirlpool. Thistheory was subsequently abandoned, but it had strongly influenced Frenchphilosophy in the person of Henri Bergson.

This was manifested notably in one of Ostwald’s works titled Energy (translatedinto French around 1910). Let us see exactly what Ostwald said:

“By energetics we mean the development of this idea that all phenomena innature must be conceived of and represented as operations carried out on diverseenergies. The possibility of such a description of nature can only be imagined oncewe have discovered the general property that all the different forms of energy possessof being able to transform into one another. Robert Mayer was, thus, the first whocould envisage this possibility” [OST 37, p. 119].


In the course of his reasoning on the “material factors of energy”, Ostwald wouldsoon arrive at the idea that it was practically possible to dispense with material inscience. “Upon analyzing matter and determining its component parts, he wrote, wehave been able to see that it constitutes a superfluous concept” [OST 37, p. 171]. Hewould then demonstrate that material systems like living systems were, in fact,“energetics systems”, and that the very concept of energy, inasmuch as it dispenseswith matter, which is only a “complexus” of energies, also did away with themind-matter opposition. The problem then became that of knowing what the relationbetween energy and the mind could be [OST 37, p. 200].

Thus, the idea of a “psychic” energy, or even “psychic energetics” , which we alsosee postulated by Freud, can be found in Bergson’s writings under a form that is barelysublimated, with the idea of a “spiritual energy” (an idea the philosopher was familiarwith) that came from Ostwald. The concept of “psychic energy” also figures explicitlyin an eponymous work, one of Bergson’s most famous works [BER 19, p. 213].

Bergson, incidentally, repeated a large part of these ideas both in Matière etMémoire (Material and Memory) and in L’Evolution Créatrice (Creative Evolution),influencing an entire generation of philosophers; there are also adherents of thisphilosophy today.

Here are two main passages we must highlight from Matière et Mémoire (Materialand Memory):

“We have no reason, for instance, for representing the atom to ourselves as a solid,rather than as liquid or gaseous, nor for picturing the reciprocal action of atoms byshocks rather than in any other way” [BER 39, p. 224].

Further, on the same page:

“The preservation of life no doubt requires that we should distinguish, in our dailyexperience, between passive things and actions effected by these things in space. Asit is useful to us to fix the seat of the thing at the precise point where we might touchit, its palpable outlines become for us its real limit, and we then see in its action asomething, I know not what, which, being altogether different, can part company withit. But since a theory of matter is an attempt to find the reality hidden beneath thesecustomary images which are entirely relative to our needs, from these images it mustfirst of all set itself free. And, indeed, we see force and matter drawing nearer togetherthe more deeply the physicist has penetrated into their effects. We see force moreand more materialized, the atom more and more idealized, the two terms convergingtoward a common limit and the universe thus recovering its continuity” [BER 39,pp. 224–225].


Further again, we find this text:

“To this conclusion we were bound to come, though they started from verydifferent positions, the two physicists of the last century who have most closelyinvestigated the constitution of matter, Lord Kelvin and Faraday. For Faraday theatom is a center of force. He means by this that the individuality of the atom consistsin the mathematical point at which the indefinite lines of force cross, radiatingthroughout space, which really constitutes it: thus each atom occupies the wholespace to which gravitation extends and all atoms are interpenetrating”.10

Bergson continues:

“Lord Kelvin, moving in another order of ideas, supposes a perfect, continuous,homogeneous and incompressible fluid, filling space: what we term an atom he makesinto a vortex ring, ever whirling in this continuity, and owing its properties to itscircular form, its existence and, consequently, its individuality to its motion"11.

And Bergson concludes with:

“But on either hypothesis, the nearer we draw to the ultimate elements of matterthe better we note the vanishing of that discontinuity which our senses perceived onthe surface”.

What do physicists think of such a construction?

P. Duhem who is also a physics historian, has a completely negative opinion onEnergetism and Atom-Vortex theory. In his landmark book The Evolution ofMechanics [DUH 92, pp. 178–179], Duhem refuses to do away with the idea ofmatter, claiming that its existence and the importance of energy in the world havebeen proven. When we hit someone with a stick, he observes, the person in questionundeniably feels the energy from the stick, but this is not to say that there is no stick!

Today, given the influence of the theory of relativity (equivalence between massand energy) and above all, the influence of the new paradigm of quantum physicsthat is superstring theory, wave-form representations, referring to a fluid universe fieldwith fields and vibrations, seem to be returning. Louis de Broglie’s remarks are, thus,even more significant now. He had refuted any theory that did not consider the entirety

10 Bergson refers here to Faraday “A speculation concerning electric conduction”, PhilosophyMagazine, 3rd series, vol. XXIV.11 Bergson refers here to [THO 67]. A hypotheses of this kind, he writes in the same note, waspostulated by Th. Graham [GRA 63, pp. 621–622].


of both the continuous and discontinuous representations of phenomena. There are, inparticular, 17 magnificent pages [BRO 37, pp. 239–256] on the reasoning of physicistsand the impossibility of contenting oneself with either continuous or discontinuousrepresentations of matter. We will discuss here just the conclusion of this text:

“The real cannot be interpreted using pure continuity: individualities must berecognized within it. But these individualities do not conform to the image that wegive ourselves pure discontinuities: they are expansive, constantly interacting and,the most surprising of it, it does not seem to be possible to localize them and definethem from a dynamic point of view, with perfect exactitude at each instant. Thisconception of individuals with slightly fluid boundaries, presented against abackground of continuity is very novel to physicists and may perhaps even seemshocking to certain among them; but does it not conform to that to whichphilosophical reflection could lead?” [BRO 37, pp. 255–256].

The use of scientific results in philosophy is, thus, a double-edged sword:Bergson, who had blind faith in energetism, saw his philosophy refuted by theevolution of physics itself. But energetism, at the time that he wrote, was a debatabletheory. He only accepted it because it seemed to go with spiritualism, of which hewas a stout defender. Philosophers, therefore, cannot be urged too strongly to carryout a solid epistemological12 study before proceeding to their philosophicalextrapolations – something that Bergson seems to have dispensed with.

12 In his youth, Bergson was rather a scientist. But his use of mathematics in philosophyremained purely metaphorical (see [MIL 74, p. 80]).

PART 3

Significant Advances



From the complex history of mathematics, which has been written and rewrittencountless times, we will only focus here on those elements that strengthen ourargument: how new discoveries “impacted” philosophy, providing it with novelmethods and powerful argumentation. Based on the needs of this thesis, therefore, wewill look at three major instances: chance entering calculations, traditional spacelosing its unicity and seeming to suddenly multiply at the same time that new(non-Euclidean) geometries emerged, and finally, the turn that modernity took, wherethe very elements that reasoning had been based on so far, suddenly became moreabstract and, losing their intrinsic substance, were now defined in greater measure bytheir relations, especially as they themselves now entered into a more and moreall-encompassing reflection. Making up “sets” and “structures”, they thus came todefine that which we once called “modern mathematics”, an expression that hasfallen into disuse today.

Mathematics was initially geometric and operated on clearly defined quantities: itcorresponded to essentially agrarian societies, where the chief problems to resolveconcerned land surveying and demarcation. The exact calculation of areas andvolumes proved to be inestimably valuable in the resolution of these problems. Withthe emergence of cities and growth in population, other problems also arose,accompanying the developments in economy and commerce. Demographic questionsentered the scene, and calculations related to interest and returns, related to investingcapital, as well as questions related to strategies and decision-making in an uncertainuniverse. It thus became unavoidable to recognize the existence of chance anddetermine this, mathematically, as far as it was possible to do so. Revealed in all itsglory in the 17th Century, this problem led to the invention of the calculus of gamesof chance, with the major innovators in this field being Huyghens and Fermat. Andthis calculus was immediately carried over into philosophy, notably by Pascal, whoserenowned bet will be examined in Chapter 7 before we go on to examine what theearly sociologists (Quételet, in particular) would have used. From this time onwards,


randomness, which would only grow in importance, entered mathematical reflection,and completely transformed the way we saw mathematics – a transformation thatendures to this day.

Two centuries later, in the 19th Century, it was geometry’s turn for dialecticism.The Euclidean model might have seemed entirely perfect but, in reality, it did containsome flaws, based on evidence that would, over time, turn out to be questionable orbased on presuppositions that were harder and harder to accept without demonstration.One of these, Euclid’s famous fifth postulate or the “postulate of parallels”, had beenthe subject of much debate from antiquity. The mathematicians of the 18th Century,starting with Lambert, were far bolder than the ancients and sought to demonstratethis postulate using a well-known method: reductio ad absurdum or argumentationusing contradiction. In accordance with C. F. Gauss, the Hungarian mathematicianJános Bolyai, was able to prove that it was possible to develop, without contradiction,an alternative to Euclidean geometry. This could be done by refuting precisely thatwhich Euclidean geometry asked us to accept: given a point outside a straight line,only one parallel to this line could be drawn through this point. The assumption thatthere could be two parallel lines passing through this point, or even several, openedthe way to hyperbolic geometry. But another form of negating the fifth postulate wasalso possible: assuming that given a point outside a straight line, no parallel line can,in reality, pass through it. Riemann who chose this second solution (which wouldalso lead to the development of a non-contradictory geometry, elliptical geometry)demonstrated, using the same method, that a given space was not necessarily flat.These significant advances would, eventually, also have philosophical impacts. Theconcept of “parallelism”, an expression Leibniz used to describe Spinoza’s system,would also become more complex and the existence of finite but non-limited spaces(which would ultimately be made tangible through the theory of relativity) would also,inevitably, have an effect on thought. The works of Clifford and then Finsler gave thisall its speculative sense.

But it was only at the end of the 19th Century that there was a genuine, qualitativeleap forward. Shaken by the now-proven existence of multiple geometries, as therewere multiple forms of algebra (associative and non-associative, commutative andnon-commutative), mathematics was forced to revisit its fundamental tenets in amore general and abstract sense. The method used by the German mathematician G.Cantor, with set theory, and then the successive works put out by Frege, Hilbert,Russell and Withehead, as well as by Peano, would usher in a new age, wheremathematical objects, growing increasingly general and abstract, would exist less bythemselves than in relation to others. A theory of “multiplicities” and structuralmathematics would be born and these two found their echo in the works of somephilosophers, either accompanied by epistemological commentaries (as withCavaillès), or through attempts at transposition to philosophy or metaphysics (as withHusserl). In all these cases, once again, mathematical advances resulted inphilosophies that are difficult to approach and that are harder still to understand


through abstraction. This was followed by a new school of thought that was moreanalytic than synthetic and which has gone from strength to strength in theanglo-saxon tradition.

This section is dedicated to retracing these three steps in the development ofmathematics and philosophy.

7

Chance, Probability and Metaphysics

Before examining how the entry of chance transformed mathematics, it would beinteresting to look at the history of this concept etymologically! There are, indeed,many different terms used for it across mathematical cultures. In English, the wordhazard appeared in Chaucer’s Canterbury Tales and has since been used as the nameof a game of chance (using dice). This is most probably derived from the French wordhasard. This is azar in Spanish and Portuguese, while in Italian it has the dual formla zara, azzardo. Several different etymologies have been attributed to “Hasard” (orchance), over the years, all of which are more or less lacking in proof. The mostplausible would be that hasard is derived from the Arabic sehar or sâr, which means“dice”. With the article al, it would have become alssahar, alssar, from which theword “hasard” could have been derived. According to Guillaume de Tyr, a chroniclerof the Crusades, alssar was, indeed, originally a game of “dice”. This was discoveredduring a siege of a fortress in Syria, which was itself called Hasart, which is whythe game was given the name of the citadel. In all these cases, it is thus hazard as adice-game that led to “hazard” in the sense of “chance”, and not the other way around[ROS 71, p. 76].

Among the terms related to “hasard”, we first have the French word sort, whichis the Latin fors, and the Greek tuchè, from tugchanô, “to obtain”. This word, “sort”in its Latin and Greek forms, is visibly an anthropological concept, designating thatwhich comes to man from good fortune or misfortune, what the gods decide is his lot.

The English word chance belongs to a different register. This is from the Latincasus, which underlies an idea that is more abstract than the earlier one as humans arenot necessarily involved. This idea can be found in several Indo-European languagesin different forms: the German Zufall, the Italian caso, the Spanish casualidad, theFrench rencontre, all referring to one and the same thing: the idea of there being apoint of intersection for multiple causal series.



Let us finally mention the more philosophical notion of contingence, from the latincum-tangere, “take or hold together”, which, consequently, is derived from the ideaof simultaneity, but which is also, in French, applied to non-necessity: in the orderof modalities, the contingent is precisely defined as that which is not necessary but ofsecondary importance.

Science, of course, cannot take all of these determinations into account. From thevast domain of the random, science only takes that is mathematizable, that whicharises from what in the 18th Century was called “the calculation of chances”.

7.1. Calculating probability: a brief history

As with a certain number of mathematical disciplines, but undoubtedly muchmore than others, the calculation of probabilities is related to concrete situations andpractices.

As M. Loève observes: “it was born as a theory for games of chance. Very soon itwas used, often with little justification, to study collective phenomena such as actuarialproblems, starting with the calculation of annuities and the analysis of mortality tables.During its classical period – through the 17th and 19th centuries, and until the ‘heroicperiod’ between 1925–1940 – the calculus of probabilities was studied for economicand social risks of all kinds, statistical predictions were made, laws were establishedfor observation errors in astronomy and others, legal problems such as the constitutionof and decisions by a jury were discussed and so on” [LOE 78]. It was only in the 20thCentury that the calculation of probabilities was liberated from its role of instrumentto becoming a distinct mathematical discipline in its own right.

The calculation of probabilities was almost immediately philosophical as, underthe name “computation of chances”, a number of questions arose concerning gamesof chance. While traces of these can be found in works as far back as those of Cardan(1525) and Galileo, it was series of old problems posed by the Chevalier of Mere toBlaise Pascal that are really behind the birth of this field. One of these is to find outhow many times dice must be rolled such that the chances of obtaining a double-sixbe greater than 50%. The other question was, how to divide the stakes in a game if thisis interrupted before the first player obtained a certain, required number of points (theproblem of points or parts).

These two problems gave rise, at least implicitly, to the concepts of probability andmathematical expectation. The same concepts can also be found in approaches usedin the same period by Huyghens (De ratiociniis in ludo aleae) and Fermat.

But the calculus of probabilities really developed as a mathematical disciplinewith the Bernoullis. Anticipating the publication of the manuscript Ars Conjectandi

Chance, Probability and Metaphysics 163

(Basel, 1713) by Jacques Bernoulli (1654-1705), in 1708 P. de Monmort publishedthe first treatise on the calculus of probabilities: Essai d’analyse sur les jeux dehasard (Essays on the analysis games of chance), in which a large number of diceand card games are analyzed using combinatorial calculations, including fiveproblems posed by Huyghens. Nicolas Bernoulli (1687–1759), Jacques’s nephew,would originate the “Saint-Petersburg paradox”, a game where the player tosses acoin for heads or tails and wins 2n francs the first time tails comes up on the nth try,with the probability p = 1/2n. The question is to find out what the player must payto participate in this game and the classic (paradoxical) response is that their price ofparticipation must equal their mathematical expectation, which is infinite. Thesophisticated distinction introduced by Daniel Bernoulli (1700–1782), Nicholas’brother, between mathematical expectation and moral expectation would persist until1937 and Feller’s work on the law or large numbers.

Jacques Bernoulli had defined the notion of a random variable1 and gave the firstversion of the law of large numbers or “Golden Theorem”. The Probability Theory,by Abraham de Moivre (1667–1754), that followed, would then generalize the use ofcombinatorics. By working more intensively with the golden theorem using theasymptotic formula for n!, he had already obtained what Poincaré would one day call“the normal law”. He correctly defined the concepts of probability, mathematicalexpectation, independence and conditional probability. He also precisely laid outrules for the addition and multiplication of probabilities. This book by De Moivre(1718) is based on the Bernoulli schema, an idealized coin-toss game, correspondingto the following situation. Given Sn, the number of realizations of an event E withprobability p in n repeated tries, the Bernoulli formula gives, for q = 1# p:

P(Sn = k) =

%n

k

&pkqn&k, k = 1, 2, ... [7.1]

In modern writing, the law of grand numbers affirms that, for every % > 0 whenn + * we have:

P(|Sn

n# p| < %) + 1 [7.2]

This is what is called even today the law of stability of frequencies.

1 Today seen as an application in the set of eventualities, the notion of random variable thussimply designates the probability values associated with the possible occurrences of a fortuitousevent.


Finally, the Central Limit theorem states that if pq $= 0, for any x ( R, whenn + *, we have:

P(Sn # np"npq

< x) + 1"2"

( x

&#e&y2/2dy [7.3]

This is the De Moivre–Laplace theorem. De Moivre posited p = 1/2, but it isLaplace who would demonstrate the general result.

This theorem would widen the scope of the central limit theorem to the sums ofuniform, independent random variables, numerous and infinitesimal. Itsdemonstration would be clarified later by Poisson (1781–1840) and Tchebychev(1821–1894), before Markov (1856–1922) made the result completely rigorous.Liapounov (1857–1918), finally, gave the central limit theorem its definitive form.

And thus concluded, at least provisionally, the vast program set off by JacquesBernoulli, Abraham de Moivre and Siméon Denis Poisson, of stating and rigorouslydemonstrating the limit theorems for the calculus of probabilities. That is, establishingthe asymptotic tendencies of natural phenomena.

Debate would still continue for some time on the relation of probabilities toapplied statistics and on the very notion of probability, with, a priori Bayes’ theorem(Bayes, 1702–1761) and Bayesians who argued that it was always possible toidentify a probability with the repetition of a series of homogenous trials and,consequently, fix the evaluation of that probability. As, moreover, certainprobabilities depend on others, the question soon arises about knowing how toevaluate these. This reflection brought forth new expressions in probability such asP(A/B), “P(A) if B”, with new laws corresponding to these, such as the law ofconditional probabilities, which we find today in the form:

P (A/B) =P (B/A)P (A)

P (B)

These laws would eventually be integrated into the classical theory of probabilities.

In the 20th Century, the calculation of probabilities was successively freed fromall concrete dependence. With the famous German monograph published byKolmogorov (1903–1987) in 1933, Foundations of the Theory of Probabilities(published in German with the title Grundbegriffe der Wahrscheinlichkeitsrechnung)the calculus of probabilities, expressed later in the language of set theory, reached thestage of axiomatization.

A probability space was then defined as the triplet (#,A, P), formed by eventover # considered as a set, the set of events A, i.e. the *-algebra defined over the


non-empty subsets of the parst of #, stable when passing to the operations of relativecomplement, countable union and countable intersection; and the probability P overA, in other words, a positive measure across the group A, such that P(#) = 1. We saythat a property will almost certainly take place in # if the set of the ( ( # where ittakes place has the probability 1.

A random variable X becomes a measurable application of a measurable space(#,A) in the real line R equipped with its Borel set S , for example from smallest*-algebra over X containing all the open sets. We thus have: X&1(S) ( A for anyBorel set S ( S . And two random variables X, X’ are equivalent if X’ = X almostcertainly.

We again say that the law L(X) or the PX distribution of a random variable X is aprobability over S , defined by:

PX(S) = P (X&1(S)) forS ( S

Starting from here, there was a reflection all through the 20th Century onstochastic phenomena. Borel’s strong law of large numbers was extended to randomvariables (Kolmogorov), and at the same time was generalized to the stationary seriesintroduced by Khintchine (1894–1959). Paul Lévy (1886–1971) and Joseph LeoDoob (1910–2004) created the powerful “martingales” tool and Paul Lévy himselfintroduced the infinitely divisible stochastic processes and carried out an insightfulstudy on Brownian movement. His work also influenced B. Mandelbrot and histheory of fractals.

With the exception of fractals, as well as the musical use of stochastic processesin the works of the Romanian-born, French-Greek composer Iannis Xenakis,philosophers did not really hear much about this powerful movement that deeplystudied probabilistic phenomena. Moreover, apart from the epistemologists in thefield, few knew of stochastic processes and structures, their tools and laws(independence, dependence, Markovian, stationarity, limit laws, normal convergence,near-certainty, etc.).

On the other hand, at the time that it emerged, the calculation of probabilitiescould be applied without major precaution to central metaphysical questions (such asthe problem of the existence of God). And then, between 1830 and 1880, it fed intothe social mathematics projects of Condorcet, Arago and others. More recently, at theother end of history, the philosophical consequences of certain scientific discoveries(notably the algorithmic theory of information, related to a redefinition of chance asincompressibility) has been able to gain some traction in philosophy, but chiefly withinthe mathematical community.

Let us return, for now, to the famous pseudo-application of the “calculation ofchance”: Pascal’s infamous “wager”.


7.2. Pascal’s “wager”

At the time that probability was emerging, mathematician-philosopher BlaisePascal (who was also a devout Christian interested in proselytizing) tried to apply thecalculation of chance to the problem of the existence of God, trying to thus force theconversion of libertines. Pascal’s text merits being presented here at length.

7.2.1. The Pensées passage

“Let us examine this point of view and declare: ‘Either God exists, or He doesnot?’ To which view shall we incline? Reason cannot decide for us one way or theother: we are separated by an infinite gulf. At the extremity of this infinite distance,a game is in progress, where either heads or tails may turn up. What will you wager?According to reason you cannot bet either way; according to reason you can defendneither proposition. So do not attribute error to those who have made a choice; foryou know nothing about it. ‘No; I will not blame them for having made this choice,but for having made one at all; for since he who calls heads and he who calls tailsare equally at fault, both are in the wrong. The right thing is not to wager at all’. Yes;but a bet must be laid. There is no option: you have joined the game. Which will youchoose, then? Since a choice has to be made, let us see which is of least momentto you. You have two things to lose, the true and the good; and two things to wager,your reason and your will, your knowledge and your happiness; and your nature hastwo things to shun, error and unhappiness. Your reason suffers no more violence inchoosing one rather than another, since you must of necessity make a choice. That isone point cleared up. But what about your happiness? Let us weigh the gain and theloss involved in wagering that God exists. Let us estimate these two probabilities; ifyou win, you win all; if you lose, you lose nothing. Wager then, without hesitation,that He does exist. ‘That is all very fine. Yes, I must wager, but maybe I am wageringtoo much’. Let us see. Since there is an equal risk of winning and of losing, if you hadonly two lives to win you might still wager; but if there were three lives to win, youwould still have to play (since you are under the necessity of playing); and being thusobliged to play, you would be imprudent not to risk your life to win three in a gamewhere there is an equal chance of winning and losing. But there is an eternity of lifeand happiness. That being so, if there were an infinity of chances of which only onewas in your favor, you would still do right to stake one to win two, and you wouldact unwisely in refusing to play one life against three, in a game where you had onlyone chance out of an infinite number, if there were an infinity of an infinitely happylife to win. But here there is an infinity of infinitely happy life to win, one chance ofwinning against a finite number of chances of losing, and what you stake is finite. Thatremoves all doubt as to choice; wherever the infinite is, and there is not an infinity ofchances of loss against the chance of winning, there are no two ways about it, all mustbe given”.


Appearing in his collection Pensées (Thoughts) and titled Infinity. Nothingness,Pascal’s text was first published in 1670. Today, the fragment Infinity. Nothingnessbears the number 418 in Lafuma’s classification and 233 in Brunschvicg’sclassification. Having been annotated and critiqued innumerable times, it deserves tobe read today in light of current developments in the calculation of probabilities.

7.2.2. The formal translation

Let us recall, in the manner of the most pertinent commentaries, that what is atstake here is not so much the existence of God, but the problem of living as thoughGod exists or living as though he does not. Having said this, Pascal’s argument leads,in modern language, to a trivial situation: we are faced with a game where we put inn, allowing us a gain N , where the probability of winning is p and the probability oflosing is 1# p. We then have the mathematical expectation:

E = p(N # n)# (1# p)n [7.4]

The quantity (N # n) is the payoff from the game. The player will be all the moreinclined to play when the mathematical expectation is large. Pascal only envisages thecase where we bet in favor of the existence of God.

Pascal’s presuppositions can be summarized as follows:

1) n , +*;

2) N = +*;

3) p > 0.

The argument is assumed to show that whatever be the probability p ' 12 that God

exists, one must bet on the fact of this existence.

7.2.3. Criticism and commentary

7.2.3.1. Laplace’s criticism

The first truly pertinent criticism from the mathematical point of view is Laplace’scriticism. This was founded on the fact that the infinite that Pascal promised was onlybased on testimony and the probability of this testimony was extremely weak. Todemonstrate this, we first replace the formula [7.4] with the following formula [7.5]:

E = p!(N # n)# (1# p!)n, [7.5]


where p! is the probability of the truth of the testimony. In other words, the probabilitythat knowing that the witness who affirmed “if God exists, they who had led a virtuouslife will have an infinity of happy life” has spoken true.

To evaluate, let us assume an urn contains a very large number, N , of balls,numbered from 1 to N . We draw a ball at random. The number of the ball we drawrepresents the number of happy lives that can be obtained in the case where weconform to God’s will, with eternal beatitude corresponding to an infinite number N .A witness is present at the draw and attests that the extracted ball bears the numberN , for example, the largest of all the numbers.

We then denote the elements as follows:

1) E is the event: the ball drawn bears the number N;

2) E' the event: the witness says that the ball drawn has the number N ;

3) P (E) and P (E') their respect probabilities, with P (E) = 1/N ;

4) V the event: the witness tells the truth;

5) F the event: the witness does not tell the truth;

6) P (V ) and P (F ) their probabilities;

7) V/E' the event V conditioned by the event E';

8) P (V/E') the probability of V if E'.

We must, therefore, calculate p! = P (V/E'). A simple calculation of conditionalprobability [PER 13, pp. 1–18] makes it possible to obtain the formula:

P (V/E') =1

1 + P (E#/F )P (F )P (E)P (V )

And assuming, with Laplace, the fact that the witness is Christian:

P (E'/F ) =N # 1

N.1

2, P (F ) =

1

10, hence : P (V ) =

9

10

we obtain:

p! = P (V/E') =18

17 +N

It then follows that when N tends to infinity p! becomes infinitesimal. Theexpectation of gain is thus:

E = p!(N # n)# (1# p!)n =18

17 +N(N # n)# (1# 18

17 +N)n [7.6]


a formula that tends to 18 # n when N tends to infinity. From this, we concludethat it is sufficient that n be strongly evaluated2 for this expectation of profit to showitself to be negative. Laplace concluded that the infinite disappeared from the productexpressing the promised advantage, which destroys Pascal’s argument.

7.2.3.2. Emile Borel’s observation

Laplace’s evaluations may, of course, be contested, but not the formula formathematical expectation [7.4], which we find in other types of formalization.Moreover, in his reasoning, Pascal does not exclude the case where there is onlychance in infinity of winning the infinite, a situation in which, according to him, it isbetter to continue playing. The problem here is that the definition of mathematicalexpectation leads us to consider the product of zero by infinity. On noting that such aproduct is, in principle, indeterminate, but that we can sometimes succeed incalculating its true value, Emil Borel [BOR 47] cited the following example, wherethe true value of mathematical expectation, in the case of an infinite gain with a zeroprobability, is zero.

We assume an infinite series of trials, in which the successive probabilities of thefavorable event are: p1, p2, ..., pn..., the series p1+p2+ ...+pn+ ... being convergentand having the sum s. The mathematical expectation of a player who must receive thegain g = 1 for each trial will, obviously, be equal to s.

It is, however, possible to evaluate this mathematical expectation in other ways.We know [BOR 09] the probabilities A0, A1, ..., An, ..., A# in order for the favorablecases to be produced, respectively, 0, 1, 2, ..., n, ...* times and we know that A# isnull. The mathematical expectation is then:

A1 + 2A2...+ nAn + ...+*A#

and an easy calculation shows that we have:

A1 + 2A2...+ nAn + ... = s

As Emile Borel remarks, “we can conclude from this that the real value of *A#is zero, that is the mathematical expectation with respect to an infinite gain that willbe realized if the favorable case occurs an infinite number of times, is zero”[BOR 47, p. 78]. The author infers from this that, in the controversy of the wager, themathematician must remain neutral and that they cannot draw a definitive argumentfor or against using science. We will see, further on, whether it is possible to gobeyond this skepticism.

2 Which would be the case, for example, with an author like Hans Jonas [JON 79].


7.2.3.3. Decision theoryBased on the von Neumann–Morgenstern method [NEU 44], we can identify

the possible life choices related to the existence or non-existence of God with a setof strategies X , over which is defined a relation of preference - that satisfies thefollowing axioms:

1) completeness: for all x, y ( X we have either x - y, or y - x;

2) transitivity: for all x, y, z ( X such that x - y, and y - z, we have x - z;

3) continuity: for all x, y, z ( X such that x - y - z, there are real quantitiesp, q ([0,1] such that: px+ (1# p)z - py + (1# p)z;

4) independence: for all x, y, z ( X and for all p ([0,1], the relation x - y isequivalent to the relation px+ (1# p)z - py + (1# p)z.

As Y. Perrin shows, “axioms (1) and (2) define a total pre-order, the axiom ofcontinuity affirms the existence of disturbances that respect the preferences, the axiomof independence signifies that the preference x - y is not modified if we mix x and yusing the same strategy z and the same probability p. The theorem makes it possibleto conclude that if the player organizes their preferences in a reasonable manner, theirwinning strategy is that which maximizes the expected utility” [PER 13, p. 11].

We thus have the matrix of gains given in Table 7.1.

God does not existGod exists (probability p) (probability (1# p))I bet on God’s existence I b

I bet against God’s existence f g

Table 7.1. Matrix of gains

For Pascal, b, f, g are finite numbers while I is an infinitely large positive.According to the von Neumann–Morgenstern theorem, the best strategy is that whichmaximizes expected utility. In the first strategy (wagering for God’s existence), theexpected utility is:

pI + (1# p)b

In the second (wagering against God’s existence), the expected utility is:

pf + (1# p)g

As the product pI is greater than any real, we necessarily have:

pf + (1# p)g ' pI + (1# p)b


The first strategy is, therefore, preferable to the second.

However, as we know, the libertine cannot be persuaded by this argument. We can,thus, use a mixed strategy that consists, for the better, of probabilizing their choices –betting for God or against God – that is subjecting their choices to a random eventwith the chosen probability q, for example, drawing a ball out of an urn, which wouldlead to an identical result. Betting for God with the probability 1 is strictly preferableto betting for God with the probability q, for any q < 1.

This presentation of the bet, adopted in many commentaries, encounters severalproblems. Thus, McClennen [McC 94] found the four axioms imposed on therelation as preferences, - (see above), and on which is based the demonstration ofthe von Neumann–Morgenstern theorem, are incompatible with the existence of autility function that takes infinitely large values.

7.2.3.4. The non-standard analysis frameworkWe suggested in 1991 [PAR 91, pp. 83–92] that the only framework that can give

meaning to the Pascal wager is that of the non-standard analysis.

In a recent article, Herzberg [HER 11] seems to have reapproached this idea byusing the non-standard theory of probability presented by Nelson [NEL 87]. Letus recall that non-standard analysis, developed by [ROB 66], makes it possible toprecisely obtain a rigorous foundation in the concepts of the infinitely large andinfinitely small. In Robinson’s presentation, the totally ordered field of real numbers Ris immersed in an extension R', which contains infinitely large elements, for examplelarger than all the real numbers in R and also infinitesimals, that is, elements whoseabsolute value is smaller than any strictly positive real number in R. Within such astructure we can finally manipulate the infinite gain I of the matrix of utilities likean ordinary number and, above all, give meaning to Pascal’s idea of infinitesimalchance (one favorable chance against an infinity of contrary chances). In particular,R' possesses several interesting properties:

1) any real p < 1 verifies pI < I whatever the infinitely large number I ;

2) if, moreover, p is not an infinitesimal, then pI is an infinitely large number and,regardless of the real number, r ( R, I + r is infinitely large.

F. Herzberg has, thus, reformulated Pascal’s wager in this non-standard model andtransferred into this model the fundamental von Neumann–Morgenstern theorem. Hewas, thus, able to show that if the order of preferences is a total preorder verifying theaxioms of continuity and independence, then it is legitimate to maximize the expectedutility, even if this is an infinitely large number. Using reasoning that is identical to theearlier reasoning, he thus endeavored to validate Pascal’s argument.

The fact remains, as we already observed in 1991, that Pascal’s infinity hasqualitative aspects as well as quantitative aspects. The philosopher surreptitiously


moves from the idea of a gain of several lives to that of “an infinity of life” (in thesingular) or even an “infinity of infinitely happy life”.

Herzberg, furthermore, does not respond to a mathematical objection raised byHàjec [HAJ 03], who believed there is no way to characterize a notion of utility thatis “reflexive for addition and strictly non-reflexive for multiplication by finite,positive probabilities” such that it does indeed seem as if the wager, at least in theform presented by Pascal, has no chance of ever being truly validated bymathematics3.

7.3. Social applications, from Condorcet to Musil

As Pierre Crépel [CRE 89], specialist in the field, recalls: between the secondhalf of the 18th Century and the second half of the 19th Century, the calculation ofprobabilities saw considerable progress in France, which drew the attention ofspecialists in the social sciences.

Condorcet, the first of these, emphasized the possibilities for applying this in thefield of the political and moral and, despite much opposition, proposed a course thatwould demonstrate how to apply this calculation not only to games of chance andlotteries, but to many other political and social subjects: mortality tables (Huyghenshad already thought of this), annuities, insurance theory, probability of testimony, etc.Laplace further extended this to problems in astronomy.

Much later, Fourier (1768–1830), with his memoirs published in the Recherchesstatistiques sur la ville de Paris et le département de la Seine, de 1821 à 1829(Statistical research on the city of Paris and the Départment of the Seine from 1821 to1829) as well as in various unpublished manuscripts, would partially return to theproject envisaged by Poisson (1781–1840). This, despite the fact that Poissonremarked that the accumulation of data, revealing certain striking regularities,showed evolutions and variations that were much harder to process, due to theinsurmountable obstacles presented by that which we call, “the variability ofchances”.

Arago, who was closer to Poisson, involved himself again in the ambitiousproject launched by Condorcet, even though the difficulties encountered, as well as

3 We will say the same of all the theories that claim to apply game theory to the behavior of aHigher Being, whether this is to bring out, and eventually see how to resolve the paradoxesof omnipresence, omnipotence or omniscience (see [BRA 00]). Similarly, philosophicalcosmologies that tend to make the idea of an “infinite spirit” plausible are not really morecredible (see [LES 01]).


the increasing formalism, prevented the development of any real philosophicalapplication. Later, Cournot4, Bravais, and Bienaymé, as P. Crépel observed, came upagainst these problems ‘more or less consciously’ and the solution to these problemswould be discovered only toward the end of the 19th Century or the early 20thCentury. Cournot’s historical etiology, which he would substitute for any philosophyor teleology of history, would not, thus, really be an “application” of the calculationof probabilities to historical science, but rather the observation that the presence ofchance and the accidental are intermingled here with reasons or necessities that wecan explain without, however, arriving at a mathematics of human evolution5.

In this respect, Belgium would prove even more daring than France. If theconcept of mathematical expectation gave rise to this debatable metaphysicalextension in the 17th Century, of Pascal’s wager, then one of the most important lawsin the calculus of probability, acquired from the 18th Century, the normal law, wouldgive rise, in Belgium, to a sociological theory which also had philosophicalconsequences. The theory called “the average man theory” was developed byLambert Adolfe Jacques Quételet (1796–1874), a Belgian mathematician,astronomer, naturalist and statistician.

In his major work, A treatise on man and the development of his faculties,Quételet shows that any society necessarily accepts a representative, typical human,“the average man”, which is the central value around which the characteristic humanmeasures are grouped in a normal curve. These characteristics are not only restrictedto the physical. Each individual is not only given a specific height and weight, theyalso have a propensity to marry, fall ill, commit suicide or homicide. The statisticaloffices are thus comparable to astronomical observatories: they record stable factsand, predicting from these, give rise to a “social measure”.

Statistics, which began developing toward the end of the 19th Century, bring outregularities, both in the census of populations as in the organizing of data fromregistries (birth, marriage, death) or criminal and sanitary statistics. Consequently,whether we wish it or not, the future of societies is subject to the influence of thesestatistical data that relegate humans or extraordinary facts into distribution tableswithout any notable impact on the course of things.

This triumph of the mean, on the social front, has undoubtedly had consequenceson Nietzsche’s theory of nihilism. It undoubtedly strengthened Nietzsche’s hatred forthe “plebeian”, the individual in the herd, or worse, his final extrapolation in the figure

4 To learn about Cournot, see Thierry Martin’s work, notably [MAR 96a, MAR 96b].5 See, in particular, the first two chapters of A.A. Cournot [COU 72].


of the ‘last man’, who was precisely the product of this leveling of values to whichevery life in society leads and which Nietzsche’s philosophy never stopped protesting.

A literary illustration can also be found in Robert Musil’s incomparable body ofwork, which includes the novel The Man without Qualities [MUS 69]. This novelpresents a human, Ulrich, whose exceptional character unendingly runs up againstconsequences of the ‘normal law’. The novelist, an engineer by training, understoodclearly that a major consequence of this law is that the ‘probable human’ wins outeach time over the exceptional human [BOU 93b]. One such fact causes Musil’s heroto become disillusioned and skeptical, renouncing all forms of activity, whethercollective (the notorious “parallel action”, a political utopia that is ridiculedthroughout the book) or even individual activity. Ulrich, for example, abandons hismathematical research quite soon, on the grounds that even if he had pursued it, itwould not have led society to any other point of evolution than that at which it arrivedanyway. In a world where heroism is drowned in the masses, extraordinary actionshave disappeared and Ulrich bitterly realizes that even if he had become a leader inhis discipline (in which he had excelled), nothing, finally “would have objectivelychanged the course of things or the evolution of the science” [MUS 69]. Whatever wemay think of such an outlook – that we may, up to a limit, qualify as a “lazyargument” given how much this social statistic determinism (a weaker form of divinepredestination) seems to resemble the position taken by the fatalists of antiquity orthe classical age – it does not lack a certain pertinence.

7.4. Chance, coincidences and omniscience

According to the theory of probability, unexpected encounters, lucky breaks ormisfortunes, the most astonishing “coincidences” (the apparently rare but significantconjunction of circumstances that we – wrongly - assume have deviated from thenatural order of things) can be perfectly explained by the law of large numbers. Theirexceptional character is mitigated by statistical neutrality. This also has its ownphilosophical consequences, beginning by challenging all theories of destiny (fromStoics to Schopenhauer) [PAR 15a].

At the same time we must recognize that the concept of “chance” or, in itsmathematical determination, that of randomness, remains partly mysterious.Although it did not completely clarify things, a breakthrough was made in the 20thCentury with the algorithmic theory of information, which attempted to clarify theidea of a random series by likening it to an algorithmically incompressible series.This would be the case with the decimal series in ", for example, or, again, any seriesof numbers which could not be generated by a computer program smaller in size thanthe one that displays the series itself.


In accordance with Greek thought, especially Platonic thought, philosophy, likescience, had earlier wagered that the universe was comprehensible, which was thesame as saying that the information it contained, regardless of how it was defined,was “compressible”. Every theory had to be economical, as Leibniz remarked in hisDiscourse on the Metaphysics (V), observing that “the decrees or hypotheses”formulated by philosophers “hold good inasmuch as they are more independent thanone another: because reason desires that we avoid a multiplication of hypotheses orprinciples, much as in astronomy we always prefer the simplest system”. It is,therefore, to our advantage to construct well-integrated theories that are greatlysimple or that have the lowest complexity (which amounts to the same thing).

In formal language, this signifies that a theory in which the information content issmaller, or whose complexity is lower than the reality it seeks to explain, is capableof rendering that reality in both the most elegant as well as the most efficient manner.The “theories of everything”, founded on quantum physics, had, until recently, thesame ambition.

This hope now seems to clash with the incompressibility or randomness of certainrealities that we encounter both in physics as well as in mathematics.

That which overtakes us or transcends us, therefore, is not another world, as webelieve - it is already here. Thus, for example, the Borel number or the ‘know-it-allnumber’ or, again, Chaitin’s famous number #.

In 1927, in a letter to the Revue de Métaphysique et de Morale [BOR 27, pp. 271–275], Emile Borel shared the idea that it was possible to inscribe, within a real number,all the responses to all the problems that could be formulated in a given language andwhich could not be answered by a simple yes or no. The list of problems in questions isinfinite but countable and it can be ordered. We can, thus, compose a number, uniqueand clearly specified, whose nth binary decimal is the response to the nth question.

Gregory Chaitin perfected this idea. His “Omega number” is defined as being theprobability that a program, drawn at random from a universal Turing machine, willhalt. If the domain of the universal Turing machine U , is called PU , that is the set ofself-delimiting programs p that can be executed by U and that halt, then by definition:

#U =$

p(PU

2&|p|

where |p| is the length of p.

This # number contains, by definition, all possible information, but its initial bitsalready enclose the information concerning a large number of interesting


mathematical problems, notably in number theory. For example, unresolvedproblems such as Goldbach’s conjecture or Riemann’s hypothesis.

Generally speaking, mathematical problems whose solution is potentiallycontained within a Chaitin Omega are problems that can be refuted in a finite time.That is, for which the program is able to halt itself after having found acounterexample within a finite time. Not all mathematical problems, however, arerefutable in a finite time. For example, knowing whether " is a normal number orwhether there exists an infinity of twin primes are not problems of this kind. And,similarly, neither is the crucial computer science question of finding out whetherP = NP . However, any number of other problems can still be implicitly used withinOmega.

For all this, however, Omega is not calculable as it is random and incompressible.Indeed, each bit in an Omega number corresponds to a simple theorem, called the“Omega N theorem”, and affirming that “the nth bit of an Omega number is 1”. Thistheorem is true or false, which signifies that the Turing machines that contribute to thedetermination of this bit either halt or do not halt. As Omega is incompressible, thesetheorems point to many independent problems.

We thus prove three results in the algorithmic theory of information:

1) there is an infinite number of independent (for instance, not being derivablefrom one another) mathematical problems (or theorems). Practically all the “OmegaN” theorems are undecidable for a formal system of the ZFC type;

2) the complexity of the theorems that can be resolved by a mathematical theory islimited by the complexity of the set of its axioms (Chaitin’s incompleteness theorem);

3) from this it follows that the only way of reducing the number of undecidableproblems is to increase the number of axioms. The number of axioms required toresolve “Omega N” theorems tends to infinity.

A related result is that all, or almost all, is undecidable in mathematics, which canbe easily demonstrated. Let us define K(S), the Kolmogorov complexity of astatement S by the number of signs in the smallest program that is able to print S.Also, let L(S) be the length of a statement S. We now evaluate the complexity of amathematical statement with the difference d(E) = K(E) # L(E). We then showthat a reasonable system of proofs can only demonstrate the statements E, whosecomplexity d(E) is lower than a certain constant k (dependent on the system inquestion). In other words, d(E) will never be large if E is demonstrable. Moreover,as J.-P. Delahaye shows, it follows quite easily from this result mentioning d that, “ifP (n) denotes the proportion of formulae of length n, true and demonstrable in areasonable system of fixed proofs S, then P (n) tends toward 0 when n tends towardinfinity. In other words: a true formula taken at random from among the formulae of


length n has a probability of P (n) of being demonstrable, which becomes null whenn tends to infinity. At infinity, all the true formulae are undecidable” [DEL 09, p. 93].

This, evidently, would not invite one to engage in or learn mathematics, nor doesit make it possible to take this discipline as a model for thought unless we know, acontrario, the power that this discipline wields in the few cases where it is highperforming. Most other techniques are, in any case, even more ineffective whenreduced to themselves. Furthermore, the arsenal of mathematical structures(algebraic, topological, order) can provide such a powerful explanation of reality thatwe can hardly hope to say anything pertinent about it without using these tools.

8

The Geometric Revolution

Until the 1830s, there was but one geometry: Euclidean geometry.

Euclidean geometry is the geometry that verifies the postulates and axioms putforth by Euclid, the Greek mathematician born in around 330 B.C. and who died in275 B.C. He was one of the three masters of Geometry in antiquity with Appoloniusand Archimedes.

We know almost nothing about Euclid except for the fact that he was born inAthens and that he eventually settled in Alexandria. The post-Classical period ofAncient Greece began around the fourth or third Century B.C. In this period, thecultural center moved from Athens to Alexandria in Egypt. This was when Ptolemy Iorganized the Museum and turned the city into the cultural capital of the Hellenicworld, attracting many dissidents from among the Greek intellectuals including anumber of Plato’s disciples who were no longer satisfied with the academy.

Euclid is, in fact, a distant disciple of Plato’s. As we know, he was the author ofone of the fundamental texts of geometry, The Elements, which spanned 13 volumesand contained all the geometric knowledge of the Ancient world (with the exceptionof conic sections) as well as number theory.

This book had ideal demonstrations that made it a school in itself over manycenturies. There are propositions that open each volume (which are either definitions,or axioms or postulates) and from which other propositions are deduced. These arecalled “theorems” and are formulated using only those elements that were providedinitially in conjunction with syllogistic reasoning (codified in Aristotle’s Analytics)and theorems that have already been proven, if any. There are no other means used.

Only the first four books of this magisterial work discuss geometry and we discussonly those books here, beginning with a brief summary.



In the first book, Euclid first introduces certain basic concepts in the form ofdefinitions, such as those of the point, the line, area or circle. He also discusses thequestion of parallelism. He then proposes five postulates considered to benon-demonstrable and five common concepts (koinai ennoiai) that he believedfollowed from common sense and that Proclus would later call “axioms” (axiomata).

The first axioms are specific to geometry: for example, the very simple statementaccording to which “given two points, there is one and only one line that passesthrough both points”.

However, there is a second, more general category of axioms, that are valid forall sciences. For instance, “if the same quantity is added to two equal quantities wethen obtain two equal quantities”. A statement of this kind holds good for numbers,segments, areas, etc.

Following these definitions, postulates and axioms, Euclid then deduces all theconventional results of geometry related to triangles, especially isosceles andequilateral triangles. He then deduces a certain number of propositions on angles and,finally, concludes with the Pythagoras theorem, providing a demonstration of this.

The entirety of Book II is inspired by the Pythagorean school of thought and thisbook uses geometric methods to demonstrate results that are today classified asalgebraic. For example, remarkable identities such as: (a + b)2 = a2 + 2ab + b2, oragain, a(b+ c+ d) = ab+ ac+ ad. Euclid gives a geometric interpretation of thesequantities: for example, according to him, the product ab is the area of a rectanglewhose sides are, respectively, a and b. Euclid also uses the letters a and b to denotequantities that he does not liken to numbers. Book II concludes with thegeneralization of the Pythagoras theorem for any triangle.

Book III brings together various properties of circles, chords, tangents andmeasurements associated with angles.

Finally, Book IV, the last book relating to geometry proper, studies polygons, howto construct polygons using ruler and compass and how to inscribe them into circles.

Of the various results discussed, many were not known to the Pythagoreans andsome of these at least are likely to be due to the Sophists.

8.1. The limits of the Euclidean demonstrative ideal

If we observe the general form of the geometry, which, along with Aristotle’s logic,makes up the demonstrative ideal that remained unchanged until the 17th Century, it isa striking fact that the rigor displayed in principle comes up against a certain numberof limitations when applied in practice.

The Geometric Revolution 181

If we examine the elements from which all is assumed to be deduced, namely thedefinitions, axioms and postulates, what is striking is the imperfections in the system.

The axioms, which may seem to be justified both by their intrinsic evidence and asconditions for intellectual activity, are still seen as incontestable. For example, axiom1, which states that “Two things that are equal to the same thing are also equal toeach other” is self-evident. Or again there is axiom 4, which in a way explains thecase of geometric quantities and is also not problematic: “Two things that coincide– that are, as we say, congruent – are equal to one another”. There is no world inwhich this could be contested. Likewise for axiom 5, which introduces considerationsof inequality: “The whole is greater than the part”. All of this is as logical as it isgeometric.

On the other hand, a lot of empiricism and suppositions slip into the definitions aswell as the postulates.

Let us first discuss the definitions. As L. Brunschwicg [BRU 81, p. 86] observes,Euclid’s definitions are nominal definitions, not real definitions. According to whatLeibniz says in New Essays on Human Understanding (Book III, Chapter 3,section 19), a real definition is that which, “shows that the thing being defined ispossible”. Euclidean definitions, on the contrary, are only nominal definitions, in thatthey are formed with the sole purpose of making the language as clear as possible,while staying close to the elementary data in the experiment.

For example, it was experienced fact for Greeks of that time that matter cannot beinfinitely divided. When dividing any material or living thing using the materials theyhad, they would soon arrive at a limit beyond which the division signified no realprogress. For a physicist or physiologist, this signified that an unbreakable point(beyond which the substance could not be divided) had been reached. The Greekscalled this unbreakable part atomos – tomos, from temnô, to cut and the prefix “a” tosignify negation. Thus, “that which cannot be cut”. The geometric entity thatcorresponds to the atom in physics is the point. Hence, we have the completelyempirical Euclidean definition of the point: “The point is that which has no part”(semeion estin, ou meros outhen) (Book I, Definition 1).

Based on this, Euclid would thus consider the point to be that element which hadno length. Similarly, he would conceive of a line as that quantity that had no breadth;“area” as being that entity that had length and breadth but no height, unlike volume,etc. Lines are then considered to be the extremities of areas and points, the extremitiesof lines. All this is heavily empirical.

Moving on to the postulates (aitèmata in Greek), we encounter a completelydifferent problem. This is that suppositions of great importance are often hiddenunder the apparent simplicity of the requirements.


For example, the first postulate requires that we accept that one line, and only oneline, can pass through two points. This means that the line is the shortest path fromone point to another and thus defines, by this very concept, the distance between twopoints. Such a statement, however, is true in a flat space. That is, to be precise, in aEuclidean space, but is not true “in itself” from a geometric point of view.

Another example: the second postulate requires that we be able to indefinitelyextend a line. Such a demand, however, can only be satisfied if we place ourselves inan infinite space and presuppose, in turn, the infinity of the underlying space.

Yet another example: postulate 3 requires that it always be possible to trace acircle of any center and any radius. This requirement seems anodyne enough. But itwould allow Euclid to then resolve the problem of constructing an equilateral triangleon a line. This construction is carried out by taking a compass, which is adjusted tomeasure the segment that forms the base. Then, place the compass at each endpoint ofthe segment and trace a circle, The point of intersection of the circles will be the thirdpoint of the triangle, which will be connected to each endpoint of the segment using aruler. However, the first question that then arises is: why must the circles necessarilyintersect? This is followed by another: assuming we have obtained this third point,who is to say that we will be able to trace the triangle? It is because we have positedpostulate 1, which assumes that only one line passes through two points that we cando this. But is this possible in all cases? If the base segment was located on a sphere,instead of being in a plane, it would not be possible to trace the triangle in questionusing a ruler and compass.

Several suppositions thus lie beneath the apparent rigor of the Euclidean deduction.The best-known include those that underpin the famous fifth postulate, or postulate of“the parallels”. As Euclid formulates it, this postulate takes the following form: “Astraight line falling on two straight lines makes the interior angles on the same sideless than two right angles, the two straight lines, if produced indefinitely, meet on thatside on which are the angles less than the two right angles”. In other words, this meansthat two lines that are not perpendicular to the same third line, must necessarily meet.The corollary to this is that two lines that are perpendicular to the same third line areparallel i.e. they will never meet. This is further expressed by saying that, “given apoint exterior to a line, it is possible to trace only one parallel to this line”.

Over time it was seen that this postulate was made up of a sort of a “theoremon the existence of parallels” and that, generally speaking, Euclidean postulates weredisguised theorems. The idea that slowly gained ground was, thus, that given the idealdeductive rigor that was initially displayed, it was inadmissible that what should beproven through demonstrations be meekly accepted as fact without demonstrations.


8.2. Contesting Euclidean geometry

This geometry has been contested since Antiquity. Geminos, one of Euclid’ssuccessors, took Plato and Aristotle as authorities in his attempt to rid geometry of allthat required verisimilitude or probability. A little later, toward the 2nd Century A.D.,one of Ptolemy’s texts, preserved by Proclus, demonstrated a methodical effort to fillin the gaps in Euclidean deduction.

In this context, the question of the fifth postulate did not wait for the ClassicalAge. From 1 B.C., Posidonius substitutes the Euclidean definition of parallels (whichwe have seen was indirect and complicated) with a much clearer definition that simplystated that two parallels are two lines that always remain equidistant.

Proclus, through whom this definition reached posterity, also indicated that itaccompanied the geometric paradox that Geminos had already noticed: there arelines that never meet but which are not equidistant. This is the case of lines that arecalled “asymptotic” to certain curves, such as the hyperbola or the conchoid.

Thereupon, an element of doubt entered: if it is possible for two lines to meetwithout, however, being parallel to each other in the Euclidean sense, then this wouldindicate that the condition Euclid lays down, namely that the lines be equidistant, isnot an essential one for the relation of parallelism. We must, therefore, find a way ofredefining the parallelism of lines in a correct manner and the most evident way ofdoing this is to demonstrate the fifth postulate.

This project would be developed toward the end of the 17th Century – 1697, to beexact – by the Jesuit priest Saccheri, an Italian logician who would strive to achievethis by using all the resources of a reasoning that was known to Euclid – that ofdisproving through absurdity.

To prove that a proposition, p, is true, we assume that its negation ¬p is true,and we demonstrate that this negation ¬p results in a contradiction, which proves theproposition p.

Saccheri begins by applying this kind of reasoning in logic, to prove certain rulesand then transposes this to geometry. However, assuming Euclid’s postulates to betrue, notably the wrong postulate of parallels, with the natural aim of being able toprove Euclidean geometry, he would also facilitate the pseudo-geometries that in away anticipated Lobatchevsky’s and Riemann’s work. It would, however, be another100 years before they entered the picture.

For example, Saccheri considered a quadrilateral ABCD, three of whose angles,we know, are right angles: +A, +B, +C. The fourth angle then, in theory, is either right orobtuse or acute. In Euclidean geometry, of course, it is necessarily a right angle. Buthow is this to be demonstrated?


Saccheri considered the two remaining hypotheses and tried to show that theyimply a contradiction. He managed this quite well for the first hypothesis (the ideathat the angle is obtuse) by showing, in a few lines, that this hypothesis leads twodistinct lines to have two common points, which is impossible according to the sixthpostulate.

For the acute angle, this is more difficult to show. All that Saccheri was able todemonstrate is that this hypothesis leads us to consider two lines that have a commonperpendicular but also a common line. This, he concluded, is contrary to the nature ofthe straight line. However, he also saw that this metaphysical declaration was nothingclose to a demonstration and encouraged his successors to go further.

In any case, through these “absurd” hypotheses, he trained mathematicians toconsider other hypotheses than Euclid’s hypothesis and to develop theirconsequences. The result, however, was that the method itself contradicted the personwho had designed it – far from always leading to a contradiction that confirmedEuclidean hypotheses, this would in fact soon result in new forms ofnon-contradictory geometry that were different from Euclidean geometry.

8.3. Bolyai’s and Lobatchevsky geometries

The truly non-Euclidean geometries1 were born around the first half of the 19thCentury, the creations of four renowned mathematicians: Gauss, Bolyaï,Lobatchevsky and Riemann.

There are, in fact, two broad types of non-Euclidean geometries that are possible:hyperbolic geometry (discovered by Gauss, Bolyaï and Lobatchevsky) and ellipticalgeometry (discovered by Riemann). Hyperbolic geometry is, essentially, the geometryof a horse saddle, while elliptical geometry is the geometry of the surface of a sphere.

Gauss was the first to discover hyperbolic geometry. However, as was often thecase, he published nothing on the subject. Consequently, it was Bolyaï andLobatchevsky who were laureled for the discovery. Who were these mathematicians?

János Bolyaï (1802-1860) was a Hungarian mathematician. The son of one ofGauss’ former classmates at Göttingen, Farkas Bolyaï, (who was professor ofmathematics, physics and chemistry), his favorite pastime consisted, in fact, of tryingto find a demonstration for Euclid’s 5th postulate.

1 For more on this, we can refer to the first chapter in H. S. Coxeter’s book [COX 98] and for amore historical point of view [ROS 88].


At the age of 13, the young Bolyaï entered the Calvinist college (where his fathertaught) having been instructed by his father up to this age. Following this, and despitehis father’s desire for him to become a scientist, he choose a military career. He studiedin Vienna from 1812 to 1822 and immediately joined the army, but had to soon leavedue to his poor health.

Given his family environment, the young János Bolyai quite naturally inherited hisfather’s passion for Euclid’s fifth postulate. Like his father, he also tried to demonstrateit, notably between 1820 and 1823.

The form in which he approached this has since become the classic form:“Through a point external to a line, there can pass one and only one parallel to thisline”. He, quite evidently, failed to demonstrate this postulate and in 1823, he gave upon this procedure and began to study the consequences of one of the possiblenegations of this postulate, namely, that given a point outside a line, there are at leasttwo parallels to this line that pass through the point.

To his surprise, the geometry that was created by adopting such an axiom isperfectly coherent.

Bolyai marveled at the coherence of the system he had obtained. He sent his workto Gauss, who responded quite discouragingly. The substance of his response was thathe had also ventured down this path and there was really nothing to gain from theseelucubrations.

The young Bolyaï, however, persisted in his course. He published his results inthe appendices to a new edition of his father’s memoirs, after which he renouncedmathematics. Even after he had abandoned his military career, he became a landowner,never returning to his earlier mathematical work.

His work would go completely unseen and he himself was never able to ascertainthe non-contradictory nature of the new geometry he had created. It was not untileight years after his death that a model of his geometry, conceived of by the Italianmathematician Eugenio Beltrami (1835–1900)2 would finally prove him right.

2 It was apparently while studying surfaces with negative curvature that Beltrami discoveredhis most famous theorem: in an article titled “Essay on the interpretation of non-Euclideangeometry” he elaborated on a concrete model of Lobatchevsky’s and Janos Bolyai’s non-Euclidean geometry and related this to Riemann’s non-Euclidean geometry. The Beltrami modelconsisted of a pseudo-sphere, a surface brought about by the revolution of a tractrix around itsasymptote. In this article, Beltrami does not explicitly demonstrate the consistency of non-Euclidean geometry but explains that the geometry developed by Bolyai and Lobatchevsky


The Russian mathematician Nikolai Lobatchevsky (1793–1856) studiedmathematics at the university of Kazan, between 1807 and 1811. He studied underMartin Bartels, one of Gauss’ students from Göttingen, who soon became a professorat the University of Kazan where he remained for the rest of his life. In 1837 he wasknighted for his services to education.

Nicolai Lobatchevsky’s fame arises from the fact that he was the first professionalmathematician to study the new hyperbolic geometry that Gauss and Bolyai hadalready considered. In a dissertation published in 1829 and titled Fundamentals ofGeometry, he developed a geometry founded on a radical negation of Euclid’s 5thpostulate, taking the following form: “Through a given point outside a line, there canpass an infinite number of lines parallel to the first”.

Based on what we know now, following the publication of Gauss’ notebooks,most of the results stated in Lobatchevsky’s dissertation were, in fact, already knownto Gauss and his entourage. We could hypothesize that Lobatchevsky learnt of thisthrough his teacher, Martin Bartels, Gauss’ student.

His work focuses on the relations between the area of a triangle and the sum ofits angles, as well as formulae to calculate the area and circumference of a circle, oragain, the area of a polygon.

Even Lobatchevsky’s work, however, had only a limited impact on themathematical community until 1868, when Beltrami explained the significance ofthese results and provided an example of the geometry that responded toLobatchevsky’s axioms.

This intervention by Beltrami gave Lobatchevsky’s legitimacy, just as it had donein the case of Bolyai. Lobatchevsky, moreover, had not been a major mathematicianbut had boldly launched a completely new sector in the field, taking the risk ofrejection.

Lobatchevsky’s hyperbolic geometry is clearly described in a text by L.Brunschvicg [BRU 81, pp. 318–319]:

“Nothing could be clearer than the progress of Lobatschevsky’s ideas, as forexample, in Pangéométrie (Pangeometry) in 1855. The definition of parallels, inconventional geometry, is insufficient to characterize a single straight line; and there

is that of the geodesics on a surface with negative curvature. The French translation ofLobatchevsky’s and Beltrami’s work, by Jules Hoüel, is accompanied by Hoüel’s proof of theindependence of the postulate of parallels. Following this, Beltrami studied other models ofnon-Euclidean geometry, such as the half-plane or the Poincaré disk.


is nothing that prevents the concept of parallel being extended to two lines thatcomprise a bundle of non-secant lines: Given a line and a point in a plane, wroteLobatschewsky, I call parallel to the given line drawn from the given point, a linepassing through the given point and which is the limit between the lines that aredrawn in the same plane, that pass through the same point and that, when extendedfrom one side of the perpendicular dropped from that point on the given line, cut thegiven line, and those that do not cut it.”

The consequences of this conception may be discussed at length with nocontradiction appearing: there is, thus, a geometry that is different from ordinarygeometry. Is this a true geometry? The answer to this question requires us to reflecton the conditions that made it possible to attribute “truth” to the proposition thatbears the hallmark of classical geometry: the sum of the angles of a rectilineartriangle is equal to the sum of two right angles. This theorem was found to bedemonstrated using only fundamental notions. That is, only data from rational or‘intuitive’ evidence. While nobody had questioned the truth of this theorem so far,Lobatchevsky said this was only because “we see absolutely no contradiction in theconsequences that we have deduced from this: and the direct measures of the anglesof right triangles are in accordance, within the limits of error for the most perfectmeasurements, with this theorem”.

The first condition was also satisfied by the system of geometry where the sum ofthe angles of a right triangle is less than that of two right angles; what remained wasthen the experiment. This, according to Lobatchevsky, could be decisive if consideredin the space of triangles whose sides were very long.

In the meantime and from a logical point of view, Euclidean geometry and thenew geometry were both retained. Lobatchevsky gave up his first idea of calling this“imaginary geometry”. This had the disadvantage of appearing, to philosophers, asthough the new science was relegated to the world of fiction; at the same time, thisname seemed to mathematicians to allude to the apparently inextricable problemsthey encountered when introducing imaginary quantities. Lobatchevsky replaced‘imaginary geometry’ with the term Pangeometry, that is, the idea of a “generalgeometric theory that includes ordinary geometry as a specific branch”.

In order to study this segment of mathematics more concretely, we will restrictourselves here to Bachelard’s study of this in his book The new Scientific Spirit, addinga little to his commentary through the effective study of the constructions used in thisgeometry.


In The New Scientific Spirit, Bachelard pursues two objectives, as he explains3

[BAC 34]:

1) to show how geometricians have managed to open the path to rationalismby going beyond the axioms of Euclidean geometry and thus creating a new spaceof rationality in place of the closed and unchangeable reasoning used in Euclideangeometry;

2) to show, in addition, the relations between these geometries and themathematical structure that forms the base of any geometry in general, which is thestructure of the group.

Bachelard’s goal was, above all, to demonstrate how progress in geometry wouldserve to dialectize the basic concepts of geometry and thus help them evolve andbe generalized. The new (hyperbolic) geometry was, in fact, founded on two novelinnovations.

Given that we have seen that nothing distinguishes asymptotes from parallels (fromthe point of view of their behavior, neither of them meets at a point), it is possible toaccept that asymptotes to a given line may be considered to be parallels. If we acceptthis comparison, then it could be remarked that given a point external to a line a, theparallels to this line d and d!, which pass through the point A are lines that separatebetween two groups of lines passing through A: those that are secant to a and thosethat are non-secant. Thus, for the line a, there are right parallels and left parallels (seeFigure 8.1).

Figure 8.1. Right parallels and left parallels

3 We set aside, here, Bachelard’s third project of commenting on Poincaré’s error in not havinggiven these non-Euclidean geometries the importance they deserved, early on.


Let us now consider Figure 8.2. In this figure, we can clearly see that the angle +Ais a right angle equal to !

2 . We know that the line b is an asymptotic parallel to theline a and we know that p = AB. The angle + is called the parallelism angle and thisangle is defined as a function of the distance p by:

+ =$( p)

Figure 8.2. The parallelism angle

We then show that this new definition of parallelism is perfectly consistent:

– if b is a parallel to a passing through A, it is then parallel to a at each of itspoints;

– parallelism is a relation of equivalence;

– parallelism also presents the following characteristics:

- two parallels are asymptotic to each other,

- if p = 0, the angle + is equal to !2 and the lines b and a are confounded.

This is parallelism of the Euclidean kind,

- if p = *, then the angle + is equal to 0,

- between p = 0 and p = *, the parallelism angle decreases from !2 to 0,

going through all the values in between.

Several effects follow from this situation:

– the first consequence is that when p is infinite, we obtain a configuration wherethe same line is parallel to two orthogonal lines (see Figure 8.3);

– a second is that, in a hyperbolic space, we can trace several parallels to the sameline. The sum of the angles of a triangle may be smaller than two right angles (Figure8.3, central figure), and, in the case where a line is parallel (that is asymptotic) to two


orthogonal lines, we can even have a right triangle where the sum of the angles isequal to !

2 (Figure 8.3, the image on the right), as the angles to infinity are null. Butwe can also come across triangles defined by three parallel lines (Figure 8.3, image onthe left).

Figure 8.3. Specific configuration

We will also see that this hyperbolic geometry makes it possible to distinguishbetween the concept of a line and a concept of a horocycle. Let us consider a bundleof parallel lines (that is asymptotic, therefore concurrent to infinity). A horocycle isa line that is perpendicular to such a bundle. If we now take a bundle of planes, andnot lines, then the surface orthogonal to such a bundle is a horosphere. Let us callthis horosphere H a plane and its horocycles lines. It is then possible to show that Hverifies the axioms of the Euclidean plane, except, of course, for that of parallels. Inother words, Euclidean geometry is valid in the horosphere. We can then say we haveconstructed a hyperbolic model of the Euclidean plane.

In other words, a “dialectic” is in operation, which presents two movements:

1) a bifurcation movement: in Euclidean geometry, the concepts of line andhorocycle, were confounded. In Lobatschevsky’s and Bolyai’s non-Euclideangeometry these concepts are differentiated. It is one thing to be a line (eventually


parallel to another) and it is another thing to be a horocycle. The concepts of planeand horosphere can also be similarly differentiated;

2) a synthetic movement: in Euclidean geometry, the concepts of line andasymptote to a line are distinct. In Lobatschevsky’s and Bolyai’s non-Euclideangeometry, these concepts are confounded. There was fusion and generalization.

Let us end by quickly discussing the second point of interest for Bachelard fornon-Euclidean geometries.

It is possible to show, in a general manner, that the geometries interpret each other.We can, thus, construct a hyperbolic model, for example, from Euclidean geometry.There is, therefore, a logical equivalence between all these geometries, which wasdemonstrated by Poincaré and which made it possible to elevate it to a very abstractgeometry concept, as it appeared, for instance, in 1899 in David Hilbert’s Foundationsof Geometry. A geometry is therefore not characterized so much by elements such aspoints, lines, planes, etc. as it is by a series of relations between these elements, whichcan finally be called by any name.

In Euclidean geometry, these relations are, in fact, displacements. Any Euclideangeometry must satisfy a set of displacement relations with a well-definedmathematical structure. This is the structure of the group. A “group” is any structuredefined over a set of elements, which provides this set with an internal associativelaw of composition, has a neutral element and for which any element is symmetric.This very general structure, defined over the set of displacements in space, is enoughto characterize Euclidean geometry.

But other geometries are associated with groups that are richer than thedisplacement groups. For instance, hyperbolic groups (for Lobatchevsky’s andBolyai’s geometries), and the Lorentz group (for Riemannian geometry), which alsocharacterizes the theory of relativity, as a result of which this physics is associatedwith a particular type of geometry.

It was, moreover, only after the new non-Euclidean geometries (such as thehyperbolic geometries) emerged that German mathematician Felix Kleinsystematically used group theory, notably in the famous Erlangen program, that helaunched in 1872 to organize all these geometries into a coherent system. And it wasby extrapolating these ideas that Sophus Lie would then propose the foundations ofthe study of continuous groups (which would, in 1884, become Lie Groups.).

8.4. Riemann’s elliptical geometry

Riemann’s geometry is more difficult to approach than Lobatchevsky’s andBolyai’s geometries, given that it is introduced as the consequence of a considerablegeneralization of the usual concepts of geometry and analysis.


Here is how L. Brunschvicg [BRU 81, pp. 319–320] describes it, as an initialapproximation:

“Pan-geometry has no place for the system where the sum of the angles of aright-triangle exceeds two right angles; Lobatschewsky even believes this has beenexcluded through a formal demonstration. This is, moreover, in accordance with whatSaccheri’s research indicated. Saccheri believed that this was the easiest to refute andthat the hypothesis of the obtuse angle would be more difficult to realize. Theabstraction effort required is something else altogether, at least for the geometrician,as what they are required to question is no longer a determined or indeterminateproperty of parallel lines but the very existence of parallel lines. A flat surface mustbe imagined, where lines can have two common points, such as the arcs of largecircles traced on a spherical surface; the contrary proposition, however, which wasintroduced as a postulate in Euclid’s Elements, has always seemed safe from thiscontestation. Riemann’s thesis Ueber die Hypothesen welche der Geometrie zuGrande liegen (1854), shows how he managed to dissociate elements that had,hitherto, been indissolubly united. Riemann starts from purely analytical concepts:he aims to construct the most general concept of space by determining the diverseforms of the metric relations that may be established between varieties (Multiplicités)of elements and which will characterize each variety. We can thus start by bringing inthe number of dimensions. This requires taking the discrimination of spatial typesfurther: Riemann’s procedure consists of considering the space in the infinitely small,instead of starting off with infinite space. He takes as his basis the element of thelinear distance, which he assumes can be expressed in the form ds =

!)dikdxjdxk

such that the problem of constituting a metric geometry can thus be posed in thefollowing form: under what condition does the measure of distance remain the same,whatever be the location at which it is carried out? Riemann resolved this byintroducing the concept of curvature of space – an original concept, undoubtedly, butone that was made possible by Gauss’ work on the curvature of different surfaces. Byapplying this concept to space, and in particular to our three-dimensional space,Riemann is in possession of the intrinsic metric relations that allow the displacementof a figure without deformation, which satisfies what Helmholtz would later call themobility axiom”.

For more on this, we must consult Riemann’s work. In the article that L.Brunschvicg mentions, “Sur les hypothèses qui servent de fondement à la géométrie”(On the hypotheses which underlie geometry), Riemann does indeed carry out aconsiderable generalization of geometric concepts, as suggested by the philosopher.

He begins by defining a general concept of a magnitude across several dimensionsthat he would call, recalling Gauss, “manifold” (varietas in Latin, Mannigfaltigkeit inGerman). Below is an extract of the central text on this question [RIE 68d]:


“Magnitude-notions are only possible where there is an antecedent general notionwhich admits different specialisations. As there exists among these specialisations acontinuous path from one to another or not, they form a continuous or discretemultiplicity4; the individual specialisations are called in the first case points, in thesecond case elements, of the multiplicity. Notions whose specialisations form adiscrete multiplicity are so common that at least in the cultivated languages anythings being given it is always possible to find a notion in which they are included.(Hence mathematicians might unhesitatingly found the theory of discrete magnitudesupon the postulate that certain given things are to be regarded as equivalent.) On theother hand, so few and far between are the occasions for forming notions whosespecialisations make up a continuous multiplicity, that the only simple notions whosespecialisations form a multiply extended multiplicity are the positions of perceivedobjects and colours. More frequent occasions for the creation and development ofthese notions occur first in the higher mathematics”.

He then explains how to determine the metrical relations in such a manifold:

– a point, here, will be defined by n variable magnitudes x1, x2, ..., xn, that arecoordinated in a space of n dimensions;

– determining a line will then be the same as whether the quantities x are given asfunctions of a variable, in other words, whether we have a function f(x).

The problem is to find the mathematical expression for the length of a line.

The simplest point of view is, then, to calculate the distance between two pointsusing the Pythagoras theorem. In two dimensions this would give us:

s2 = (x1)2 + (x2)

2

and the theorem can be generalized to all dimensions. Thus, we have in general:

s2 = (x1)2 + (x2)

2 + ...+ (xn)2

The length is the square root of this expression, that is, an infinitesimal form. Andby using the summation sign:

ds ="$

(dxi)2

One such expression is:

4 See the following works by Gauss, Theoria res. biquadr., Book II and Anzeige zu derselben(Werke, Book II, p. 110, 116 and 118), cited in [RIE 68d].


1) homogenous to the degree 1;

2) such that f(x) = f(#x) (even function);

3) increasing monotone;

4) such that min f(x) = f(0) (zero is a minimum);

5) such that f !(0) = 0 (the first derivative is null at the zero point);

6) such that f”(x) > 0 (the second derivative is positive at all points).

From this, Riemann defines the concept of curvature of a manifold. Planemanifolds have null curvature. The manifolds with constant curvature aregeneralizations of manifolds with null curvature. Riemann’s elliptical geometry isdefined over manifolds with constant positive curvature for example, the sphere, intwo dimensions.

8.5. Bachelard and the philosophy of “non”

What meaning do we give to the negation in the expression “non-Euclidean”?

1) One possible interpretation would consist of accepting that non-Euclideangeometries are geometries that contest Euclidean geometries and also, by the sametoken, geometries that are possible and which must be placed on the same level. Thisis Poincare’s typically conventialist position, for instance. Bachelard makes thefollowing statement [BAC 73, pp. 40–41]:

“When Poincaré demonstrated the logical equivalences between the differentgeometries, he affirmed that Euclid’s geometry would always remain the mostconvenient and that in the case of this geometry conflicting with physics experiments,it would always be preferable to modify the theory in physics than to changeelementary geometry. Thus, Gauss had claimed to astronomically test anon-Euclidean geometry theorem: he asked if a triangle traced between stars, andconsequently with a vast area, would manifest the decrease in area indicated byLobatchevskian geometry. Poincaré would not admit the crucial character of such anexperiment. If it is successful, he said, we would then decide that the light ray hadsuffered some physical disturbance and was no longer propagated as a straight line.In any case, we would preserve Euclidean geometry”.


This is what Bachelard believed:

“This opinion appears to be more than a partial error and we find in it more thanone indicator for caution in the previsions of the destiny of human reason. And byrectifying it, we arrive at a veritable overturn of value in the rational domain and wesee the primordial role of abstract knowledge in contemporary physics”.

2) A second interpretation of this negation consists, in effect, of seeing that themeaning of the “non” in non-Euclidean geometries is that of a generalization, as hasalready been suggested. This second interpretation is based first of all on followinggeometry texts:

a) the conclusion of Brunschvicg’s first text showed that Lobatchevsky, withhis Pangeometry, had substituted conventional geometry with the idea of “a generalgeometric theory” which included ordinary geometry as a particular case;

b) similarly, Riemann established that “the varieties for which the curvaturemeasure is equal to 0 throughout may be considered as particular cases of varieties forwhich the curvature measure is constant” [RIE 68d, p. 292].

The “non” here is equal to a generalization, as Bachelard saw it, citing themathematician Houël in The New Scientific Spirit. Such an interpretation then makesit possible to break away from conventionalism (commodism), which are forms ofrelativism [BAC 34, pp. 30–31]:

“The Euclidiands believe(d) that their geometry was being negated, although therewas only a generalization and Lobatchevsky and Euclid could indeed agree on manythings. Generalized geometry is a method analogous to that followed by an analystwho, having just found the general integral of the differential equation to a problem,would discuss this integrant before making the constant distinct based on the data ofthis problem. This, in no way, would negate the fact that the arbitrary constant mustfinally be given such or such a value”.

But generalization goes hand in hand with completeness and coherence, “the onlypossible basis for realism”, Bachelard states [BAC 34, p. 32]. In reality, the reason wemust reject Poincaré’s conventionalism is because of the solidarity that exists betweennon-Euclidean geometry and contemporary physics.

What underlies all geometries, as we have recalled, is the algebraic structure of thegroup. The group associated with Euclidean geometry is thus the displacements group,which is relatively simple and forms the basis for the laws of Galilean physics. ButBachelard shows that non-Euclidean geometries that are associated with richer groupsare also more apt to describe a more subtle physical experiment. Thus, the groupthat underlines Riemannian geometry is the Lorentez group on which the restrictedtheory of relativity is based. Riemannian geometry is thus the geometry of the theory


of relativity and we have no choice here if we wish to take into account the physicalexperiment5.

These words result from the generalizations that entered geometry in the secondhalf of the 19th Century and at the beginning of the 20th Century. These were verywell captured by Felix Klein’s Erlangen program.

8.6. The unification of Geometry by Beltrami and Klein

Non-Euclidean geometries appeared in history in a somewhat empirical manner,through the question of the demonstration of the postulate of parallels. But a largemovement to unify geometry began in the second half of the 19th Century withBeltrami, first, and then with F. Klein and his famous Erlangen program in the 1870s.A text by F. Russo summarizes the gist of this history, noting that while the Englishmathematician A. Cayley did indeed play a fundamental role in the elaboration of theformalisms associated with projective geometry, it was Beltrami who first had theidea of unifying non-Euclidean geometries [RUS 74, pp. 57–58] :

“It is, it would seem, that it was Beltrami who was the first to highlight the commonnature of the two geometries, that of Bolyai-Lobachevsky and that of Riemann. In1868, in Saggio d’una interpretazione della geometria non euclidea, he showed thatit was possible to consider that the Bolyai-Lobatchevsky geometry was, in the caseof two dimensions, which are equivalent to the geometry on a surface of negativecurvature, the same as Riemann geometry being a geometry on a surface of positivecurvature. We know that this surface with negative curvature, which he called thepseudo-sphere, was not a completely appropriate image of the Bolyai-Lobatchevskygeometry. Nonetheless, this ‘model’ opened the way to the constitution of a generaldoctrine of non-Euclidean geometries.

Contrary to what the expression “Cayley’s geometries” would suggest, althoughthis is used quite commonly to designate the Bolyai-Lobatchevsky and Riemanngeometries, which are seen as particular projective geometries, the credit for thissynthesis does not go to Cayley who, as we have said, did not bother to work ingreater detail on the nature of non-Euclidean geometries; moreover, he makes noreference to this in his Mémoire of 1859.

5 We must also recognize that this position of Bachelard’s is possibly a bit excessive. Othergeometries, for example, Finsler’s geometries, could also agree with the theory of relativity.But the other geometries are, it must be admitted, themselves variants or even generalizationsof Riemannian geometry.


It does indeed seem that it was Klein who was the first to highlight the projectivenature of non-Euclidean geometries by applying Cayley’s views to these. Klein wasalso the first to clearly elucidate the concept of a projective measure. Cayley’sdevelopment of this subject was incomplete with respect to several aspects.

Klein clearly established that the three types of geometries (Euclid’s,Bolyai-Lobatchevsky and Riemann’s were particular cases of Cayley’s generalmetric. Posing the general problem of determining projective geometries withconstant curvature, he showed that there could only exist three types whichcorresponded exactly to these three geometries.

Klein, quite rightly, highlighted this remarkable fact that these three geometries(Euclid’s, Bolyai-Lobatchevsky and Riemann’s) thus found themselves definedthrough considerations that were completely different from those through which theyhad been introduced. Moreover, Klein was the first to show that projective geometryis independent of the theory of parallels. We know that neither Poncelet, nor Chasles,nor Staudt, nor Cayley elucidated this point and that they did not even really pose thisquestion.

And now see Klein’s own account that spoke of his procedure and how it wasrelated to non-Euclidean geometries [KLE 74, note V, pp. 40–41]:

“The projective metrical geometry alluded to in the text is essentially coincident,as recent investigations have shown, with the metrical geometry which can bedeveloped under non-acceptance of the axiom of parallels, and is today under thename of non-Euclidean geometry widely treated and discussed. The reason why thisname has not been mentioned at all in the text is closely related to the expositionsgiven in the preceding note. With the name non-Euclidean geometry have beenassociated a multitude of non-mathematical ideas, which have been as zealouslycherished by some as resolutely rejected by others, but with which our purelymathematical considerations have nothing to do whatever. A wish to contributetoward clearer ideas in this matter has occasioned the following explanations. Theinvestigations referred to on the theory of parallels, with the results growing out ofthem, have a definite value for mathematics from two points of views. They show, inthe first place – and this function of theirs may be regarded as concluded once forall – that the axiom of parallels is not a mathematical consequence of the otheraxioms usually assumed, but the expression of an essentially new principle of spaceperception, which has not been touched upon in the foregoing investigations. Similarinvestigations could and should be performed with regard to every axiom (and notalone in geometry); an insight would thus be obtained into the mutual relation of theaxioms. But, in the second place, these investigations have given us an importantmathematical idea – the idea of a multiplicity of constant curvature. This idea is veryintimately connected, as has already been remarked and in section 10 of the textdiscussed more in detail, with the projective measurement which has arisen


independently of any theory of parallels. Not only is the study of this measurement initself of great mathematical interest, admitting of numerous applications, but it hasthe additional feature of including the measurement given in geometry as a special(limiting) case and of teaching us how to regard the latter from a broader point ofview. Quite independent of the views set forth is the question, what reasons supportthe axiom on parallels, i.e., whether we should regard it as absolutely given, as someclaim, or only as approximately proved by experience, as others say. Should there bereasons for assuming the latter position, the mathematical investigations referred toafford us then immediately the means for constructing a more exact geometry. Butthe inquiry is evidently a philosophical one and concerns the most generalfoundations of our understanding. The mathematician as such is not concerned withthis inquiry, and does not wish his investigations to be regarded as dependent on theanswer given to the question from the one or the other point of view”6.

Thus, far from according the trend of non-Euclidean geometries the least bit ofinterest in themselves, Klein considered them to be particular cases of projectivegeometries which are, like all geometries, linked to the invariance of a certain group.For example, the Lorentz group in the case of Riemannian manifolds. As forknowing which is the true geometry of the real world and whether the axiom ofparallels from Euclidean geometry does or does not possess an absolute value, Kleinbelieves that this question goes beyond mathematics and is a philosophical questionthat has not been debated, let alone settled. We can even question, in truth, whetherthis is a philosophical question or, at any rate, an interesting one. All that we can sayis that physics needs a geometry that is appropriate for its model and that thisphysico-geometric complex is what defines what we call reality – physical reality atany rate – which is influenced and refined over history and based on the models weuse.

8.7. Hilbert’s axiomatization

Moving on to Hilbert’s reflection on the Fundamentals of Geometry, this wouldmake it possible to characterize geometric elements in a completely general andformal way. This is the ultimate stage of generalization: presenting geometry in anaxiomatic form would only retain essential language and non-anecdotal elementswhich would allow the geometry to develop.

6 See F. Klein, “A Comparative Review of Recent Researches In Geometry (Progamme onentering the Philosophical Faculty and the Senate of the University of Erlangen in 1872)”, NoteV, “On the so-called Non-Euclidean Geometry”, Translated by Dr. M. W. Haskell, AssistantProfessor of Mathematics in the University of California. Published in Bull. New York Math.Soc. 2, (1892–1893), 215–249.


According to Hilbert, the set of geometry can be summarized in 27 axioms, dividedinto six groups: axioms of incidence, axioms of order, axioms of parallels, axioms ofcongruence, axioms of continuity and axioms of completeness.

As was recalled with some amusement [WAR 01, p. 188], Hilbert often said thathis axioms related to concepts close to those that nature offered us, but that we couldalso replace “point” by “beer mug”, “line” by “chair” and “plane” by “table” withoutany injury to the purely logical nature of the axiom.

Even without this, there is a striking difference between Hilbert’s axioms andEuclid’s.

For example, an axiom such as H14, often called the “Pasch axiom” after MoritzPasch (1843–1930), is an axiom that translates the convexity of a triangular plaque.This was completely unknown to the Greeks and their successors even though it is asilent presence in many of Euclid’s demonstrations in Elements.

This is the same, moreover, for practically all the axioms of order, because thequestions of topology (in the sense of Analysis situs) would take a long time to beconsidered important.

The division of the axioms into their groups is also interesting. It makes it possibleto properly arrange them, which is essential for such or such a type of property. Wefind an indirect reflection of this in Euclid’s work in the fact that he delayed, as faras possible, over 28 propositions, the moment when he would use his famous fifthpostulate.

The axioms of incidence are sometimes called “axioms of belonging”. The lastgroup (the axiom of completeness) is sometimes encompassed within the earlier one,with the famous axiom, known as the Archimedes (H26). One of Hilbert’s triumphswas that he showed the existence of non-Archimedean geometries, where only thisaxiom was inexact, a real tour de force.

The axioms of congruence correspond, practically, to five (or nine, or ten)Euclidean axioms. We find, on two occasions, the characterizations of relations oforder that would reappear in modern mathematics.

Even if Hilbert’s axioms are not ordinarily used by the geometrician, the fact isthat they exist and remain perfectly compatible with our requirement of rigor, over acentury after they were penned.

As such, they mark, along with their famous Greek predecessors, an essential pointin the history of thought.


8.8. The reception of non-Euclidean geometries

Poincaré’s case clearly showed that non-Euclidean geometries had been difficultto accept. In fact, a number of eminent scientists in the 19th Century resisted thesedevelopments. Among them were A. de Morgan, L. Carroll, G. Frege and A. Cayley.Among the philosophers, there were Delboeuf, Renouvier and Stallo (the Americanepistemologist). They were so mistaken in their considerations that the position ofAristotle (who does not seem to have ever judged certain statements of Euclideangeometry demonstrable, nor explicitly rejected the inverse affirmations) seems almostmodern. Imre Toth, in an article that was already old, [TOT 77], highlighted thenecessity of accepting the multiplicity of geometries, without seeing, at the sametime, that this proof of completeness went well beyond conventionalism. Not onlydid geometry progress in the direction of its fundamentals, but the new geometriesjoined up with new physics, thus creating a new scientific complex in which physicsand geometry were indissociable. Among the philosophers, Bachelard thus remainedthe one who undoubtedly had the deepest understanding of the importance of such arevolution.

8.9. A distant impact: Finsler’s philosophy

On the epistemological level, philosophers have sometimes commented onnon-Euclidean geometries (this was the case with Bachelard) but in general, they didnot measure the importance of these new models for philosophy itself and, lackingthe competence, were unable to use them. As we highlight elsewhere, from a certaintime onwards, it was the remaining mathematicians who were best placed to perceivethe philosophical importance of their discoveries. Paul Finsler (1894–1970),Riemann’s student, and the originator himself of a type of manifold that was differentfrom Riemannian manifolds (“Finsler spaces”) was able to draw certainphilosophical considerations from these models, which had, so far, been quiteignored by the philosophers.

Finsler’s geometry is a type of Riemannian geometry, even if it is different from theusual. Given an n-dimensional manifold , where the distance between two neighboringpoints is given by the formula:

ds = F (xi, dxi), i = 1, 2, ..., n

we only require that the function be homogenous to the degree 1 in the dxi. In otherwords:

F (xi,)dxi) = )F (xi, dxi), ) >0 [8.1]


On the other hand, we no longer require that F be the square root of a quadraticform, with the result that Finslerian geometry becomes that of the simple integral:

I =

( b

aF (xi,

dxi

dt)dt [8.2]

In a text written toward the end of his career, for an orphanage in Zürich [FIN 58],Finsler launches into a very interesting metaphysical construction implicitly based onhis earlier work. Reducing the notion of “life after death” to the life of others, butalso considering that all human lives are interlined and in reality constitute only asingle life, he imagined a “Weiterleben”, an indefinite continuation of human lifemade up of the sum of all individual lives. This, in turn, assumed a sort of “time of alltimes”, the sum of all the subjective temporalities, which is a weakening of the ideaof eternity, a human eternity. From here, if all human life is already comparable to asimple integral over a Finsler space, there is a sum of all these integrals over atangent space which makes it possible to bring together subjective space times.Finsler thus went much further than Bergson: individual durations are no longersimple, heterogeneous, qualitative multiplicities that are experienced, but areauthentic Riemannian manifolds, which, lacking the power to directly connect toeach other (it is not possible, in general, to establish a “connection” betweenRiemannian manifolds) would connect to each other via their tangent space. All thelives thus form a single life, which immediately made visible human unity and itsindissoluble solidarity with the isolated (orphans) or the excluded.

But the benefits of Riemannian geometry is also that it introduces a distinctionthat philosophy does not really know, despite Kant, which is the idea of limit and theidea of finitude. The new geometry would make it possible to think of the existence ofa finite and yet unlimited space, the novelty of which was already being highlightedby Einstein. While death is an empirical end of all individual human life, it doesnot follow from this that life is limited. Though Western philosophy has, from thebeginning of modernity, unceasingly meditated on the question of human finitude,Finsler breathed new life into a speculative meditation that could then sustain itselfdespite this limit, without falling back onto this “innocent” use of the idea of infinityas it was seen by the Classic Age, but as it was denounced by Merleau-Ponty. Thisis one of the most direct impacts that we can note of non-Euclidean geometry on themetaphysical7.

7 For more on these questions, see our commentary [PAR 99, pp. 136–137 and 174–182].

9

Fundamental Sets and Structures

We can, undoubtedly, see the beginning of modern mathematics in GeorgCantor’s set theory. The term “set” (Menge, in German) was first used in von Staudt’sGeometrie der Lage (1856) but it was Cantor who would accord it the importance itnow has. The terms Menge and Mengenlehre appear briefly in a note appended to an1883 article [CAN 83a, pp. 545–591], but as G. Dauben [DAU 90, p. 170] reports,they were not definitively adopted until much later [CAN 95]. The Englishexpression set theory appeared toward the mid-1920s [FRI 26, p. 487], the mostprecise name being axiomatic set theory, which appeared in the 1930s, while naiveset theory, which was occasionally used in the 1940s, would only become anestablished term 10 years later1.

9.1. Controversies surrounding the infinitely large

Cantor’s theory commenced with considerations on the infinitely large which mustbe given its place in history.

Since the time of Aristotle and the distinction between potential infinity and actualinfinity, schools of philosophy were split into two camps. One camp only recognizedthat the concept of potential infinity is pertinent, while the other did not hesitate toaccept the effective existence of an actual infinity.

Most philosophers have traditionally displayed great distrust toward actual infinity,that is an infinity given in an immediate and completely determined manner. How can

1 It appeared in a commentary by Hermann Weyl on a work edited by P. A. Schilpp [SCH 46,p. 210] and, simultaneously, in Laszlo Kalmar’s analyses of the Kleene and Rosser paradoxes[KAL 46, p. 136].



an infinity, which has no limits by definition, be defined in any way, in other words,be de-finite in the etymological sense of the word? The infinite can, in principle,designate only the possibility of going further in a numerical series, but it cannotcharacterize an entity, that is a determined number.

There were certainly two parties present, and in every century, the debatecontinued just as fiercely.

In the 17th Century, for example, and in opposition to Descartes, Spinoza haddeclared that there was an actual infinity. In addition, he distinguished betweengeometric infinity, which he believed was the very model of nature’s infinity, andarithmetic infinity, related to the series of whole numbers. However, he found this tobe without value as it was associated only to an “imaginative” representation of theworld. This distinction, which neither Pascal nor Leibniz knew about, would be takenup again in the 19th Century by Hegel, with the contrast of “good” infinity and “bad”infinity, in the “Observation on the Infinite” in his Science of Logic, a doctrine onbeing.

In the 17th Century, Bernoulli and Fontenelle were strong supporters of actualinfinity, and in the 19th Century, Wronski and Cournot also became flag-bearers of thisidea. On the other hand, Buffon and Wolf were defenders of the Finitist arguments andRenouvier, one of the philosophers most hostile to the idea of actual infinity, stressedin his book Les labyrinthes de la Métaphysique (The Labyrinths of the Metaphysical)the “mystical” character of this infinity and the need to banish it completely from thefield of mathematics.

As concerns the thinkers, they often highlighted the paradoxical nature of infinity.From the arithmetic point of view, the idea of “larger than all numbers” wascontradictory as it was always possible to provide a number that was larger than agiven number. Similarly, from the geometric point of view, the idea of actual infinitedivisibility of a line segment also seemed irrelevant, given that it was always possibleto divide an interval given the range. Thus, all speculation on the concept ofmathematical infinity seemed, a priori, disastrous.

From Galileo to Bolzano, in particular, the paradoxes of infinity have beenhighlighted several times.

In his Discorsi of 1638, Galileo was already reflecting on the fact that if we acceptthat a line segment can contain an infinity of points, then the existence of a largersegment is the same as accepting that there existed quantities that were infinitely largerthan one another. Moreover, he suggested, at this time, that it was probably essential toidentify these infinities, which contradicted his first assertion. Similarly, in arithmetic,Salviati, one of the characters in Discorsi, remarked that if we accepted the actualinfinity of whole numbers, then we must also accept the infinity of squared numbers

Fundamental Sets and Structures 205

or, again, of roots and, as in geometry, identify these infinities. All of this seemed tosuggest, as Galileo remarked, that we could not apply the same rules to infinity asthose applied to the finite.

In a letter written in 1831 to Heinrich Schmacher, Gauss again proposed “the mostvociferous objections to the use of an infinite magnitude as if it were complete”, ausage that, according to him, was never allowed in mathematics as infinity and, in hiseyes, was only a manner of speaking that, in truth, essentially helped designate limits.

The aporetic nature of infinity was also highlighted by B. Bolzano at the beginningof his book on Les paradoxes de l’infini (The Paradoxes of the Infinite) [BOL 89,p. 1]2. The mathematician noted here that “most of the assertions that we come upagainst in the field of mathematics are made up of affirmations that contain the ideaof the infinite in a direct manner or, again, through some procedure, are based on thisidea in their argumentation”.

In short, mathematicians were generally opposed to the idea of an actual infinityand, as neatly summarized by Jean Cavaillès (who had dedicated his dissertationRemarques sur la formation abstraite de la théorie des ensembles [Remarks on theAbstract Formation of the Set Theory] to this question), mathematics of the 19thCentury effectively opened up over a “crisis of the infinite” [CAV 62, p. 31].

In Cantor’s time, toward the end of the 19th Century, this crisis was still unresolvedand the most virulent mathematician with respect to this was undoubtedly Kroneckerwho, in 1877, went as far as banning the publication of an article by the founder ofthe set theory. It was not until Dedekind intervened that it was possible to go beyondthis. But Kronecker was not the only one to manifest hostility toward Cantor’s ideas.About 8 years later, in 1885, when Cantor wanted to publish a series of articles on thenew types of order in Acta Mathematica, the Swedish mathematician, Mittag-Leffler,the founder of this review, stonewalled him.

As we can see, Cantor came up against very strong resistance from themathematical community, in order to bring about what Hilbert later called “theparadise of thought” but which he must have experienced as veritable hell at the time.

Cantor was all the more shaken in his research as, not knowing how to justify histheory, he looked for arguments from philosophy and theology. Moreover, theologiansthemselves had excellent reasons for rejection of actual mathematical infinity as theonly actual infinity that they could accept was God.

2 And on the paradoxes of the infinite in general, see [DEL 52].


The historical authorities of the Church were also divided. Thomas Aquinas, inparticular, stated his formal opposition to such a concept in his Summa Theologica.In (P1, Q7, A4), he posited various arguments in favor of actual infinity, which hethen hastened to refute one after the other. In a later work, however, De aeternitatemundi contra murmurantes, he wrote: “Until now, it has not been demonstrated thatGod cannot create an infinite multitude in actuality ... actual infinity does notrepudiate God’s absolute power, as it does not imply any contradiction”. We cannotthus absolutely consider that St. Thomas had pronounced against actual infinity.

However, this did not prevent the Church from hunting out the rare philosopherwho was bold enough to defend the existence of infinity. For example, GiordanoBruno, who was, as we know, burnt at the stake in Rome in 1600.

It is true that St. Augustin, who is conscientiously cited by Cantor, had affirmedthat God knows (or could know) infinity. But it was one thing to affirm this and anotherthing to say that actual infinity existed in the world and that man could understand itintuitively and define it formally.

Toward the end of the 19th Century, the theological dispute suddenly rebounded,under the influence of Pope Leo XIII and his important encyclical Aeterni Patris, inwhich he asked Church and Science to draw closer, though not to update theologywith modern knowledge as much as to show scientists “what must be done forreconciliation with the true principles of Christian philosophy”.

Cantor, who was the target of several mathematicians, then tried to base himselfon the Church. The problem was that he was Platonic, hence realistic, and heconsequently thought that numbers had a concrete ontological existence, including,of course, the infinite numbers that he created.

A dispute then followed within the Church between a representative ofneo-Thomism, P. Constantin Guberlet, and another German theologist, P. CasparIsenkrahe, the latter continuing to state that actual infinity was contradictory.

P. Guberlet, in an 1886 article, which returned to Cantor’s set theory, defendedthe idea that God had real awareness of a complete infinity. Thus, it followed that theideal existence of a determined value for " must be accepted, with an infinite numberof decimals and, generally, any mathematical conceptualization of actual infinity suchas Cantor’s work.

However, P. Guberlet veered away from Cantor himself on a major point: heaccepted the idea of the immanent reality of the actual infinity, but denied it any“transient” value, that is any concrete existence in nature.

This attitude was shared by the great Jesuit theologist of the time, CardinalJohannes Franzelin, who discerned the most unfortunate perversions in Cantor’s


doctrine: in effect, according to him, believing in the existence of an actual infinity,“in natured nature” (natura naturata), led directly to the pantheist error. Thistheologist believed that only God could possess true infinity.

Cantor accepted he was in the wrong and, in a letter to the Cardinal dated January22, 1896, he explained that there was, effectively, infinity and infinity. On aconciliatory note, he agreed to distinguish between an infinitum aeternum increatumsive absolutum (an eternal, uncreated and absolute infinity) and an infinitum creatumsive transfinitum (a created or transfinite infinity), the first belonging only to God andthe second being solely that of creatures 3.

This was therefore the philosophical and theological context of Cantor’s creation.Cantor, who was Christian by education, could not introduce infinity into mathematicswithout referring to the Church. But this creation now survives by itself and has rapidlybecome completely independent of the non-mathematical context that presided overits apparition. Let us now see what it consists of.

9.2. The concept of “the power of a set”

It was in 1882 (in the context of a reflection on the theory of sets of points, fromwhich would emerge the theory of abstract sets) that Cantor first mathematicallydefined the possibility of this actual infinity, which he would then baptize“transfinite”4, to keep the peace with the authorities.

In a very curious manner, the rigorous mathematical definition of this actualinfinity (or transfinity) would go through an initial redefinition of the concept ofpower. This old, Aristotelian concept, close to the concept of virtuality, and finally,quite obscure from a rational point of view, would actually constitute a sort of“epistemological” obstacle to any veritable work on infinity.

Cantor, thus, first offers a new definition of this: a set being a collection ofobjects that possess one or more common properties, we will call the possibility ofthe elements of this set biunivocally corresponding with those of another set, thepower of a set. Two sets are said to “have the same power” if we can establish abijection of one over the other between their respective elements.

3 We can, here, refer to J. Dauben [DAU 77, DAU 90]. Also see the article by P. Thuillier whowas inspired by this [THU 77].4 Cantor’s principal texts on the subject are [CAN 83b], [CAN 84], [CAN 92], [CAN 95] and[CAN 32].


This possibility of establishing biunivocal correspondence between sets of thesame power would result in a veritable classification of infinite cardinals.

9.2.1. The “countable” and the “continuous”

Cantor would first show that two classes of sets could be distinguished: those thatare equivalent in power to the set of natural whole numbers N (and which we call“countable”) and those that are equivalent in power to the set of real numbers R (andwhich we say have the power of the “continuous”).

Among the countable sets, we have not only N, the set of natural positive or nullwhole numbers, but also the set of relative whole numbers (which is today called Z);there is also the set of even numbers, the set of odd numbers, the set of squares, cubes,etc., and even, as we will see later, the set of rational numbers Q.

These sets can be made equivalent from the point of view of their “power”. Ineffect, let:

N = {1, 2, 3, 4, 5, ..., n, ...}

Inasmuch as a given whole number always corresponds to another doubled numberwe can easily make N correspond biunivocally to the set of even numbers P , that isthe set:

P = {2, 4, 6, 8, 10, ..., 2n, ...}

Similarly, the set I of odd numbers I = {1, 3, 5, 7, 9, ..., 2n+ 1, ...} will have thesame power as N, and also the series of squares, C:

C = {12, 22, 32, 42, 52, ..., n2, ...}

These results may naturally appear paradoxical as they imply identifying a set withone of its parties: for example the set N and the set of its even numbers. But this isexactly the case, and that is how good Cantor’s definition of an infinite set is: a set issaid to be infinite if it is equivalent to an integral part of itself. It is said to be finite inthe contrary case.

Thus, what initially makes up an unsolvable paradox, now becomes a foolproofmethod of distinguishing a finite set from an infinite set and precisely characterizingthe latter.

Cantor called the power of N and the sets that are equivalent to it, the power of thecountable. It is denoted by the first letter of the Hebrew alphabet (.) and a zero index.The countable is thus called .0 (aleph zero).


The question then arises again of knowing whether N has the same power as otherwell-known sets. For instance, the set that is today denoted by the letter Q and is theset of rationals, or again the set that is today denoted by the letter R and is the set ofall the real numbers.

For Q, the response is positive. To prove that this set has the same power as N,Cantor considers all fractions of the type p

q the sum of whose terms is equal to awhole number n, in other words, all the solutions, in positive whole numbers, of theequation:

p+ q = n

We can also easily show that there are n # 1 solution fractions that are, indecreasing order of magnitude:

n# 1

1,n# 2

2, ...,

2

n# 2,

1

n# 1

By successively making n equal to 2, 3, 4, etc., we obtain successive groups of 1,2, 3, etc., fractions. If we arrange all these series following one another, we form aseries of fractions that are similar to the series of whole numbers, that is a countableseries:

1

1,2

1,1

2,3

1,2

2,1

3,4

1,3

2,2

3,1

4, ...

1 2 3 4 5 6 7 8 9 10, ...

As shown by Louis Couturat [COU 02, p. 622] (from whom we borrowed theabove demonstration), we can also use an analogous demonstration to prove that theset of real algebraic numbers, in other words, the set of numbers that are the solutionsto an algebraic equation with whole coefficients, positive or negative, of degree n, iscountable.

However, as we will see, this is not the same with the set of all real numbers, R.

9.2.2. The uniqueness of the continuum

To prove that the set of real numbers contain many more numbers than the set ofwhole numbers N, Cantor works through two steps:

1) he first shows that the set of real numbers has the same power as the set of realnumbers comprised between 0 and 1;

2) he then shows that there are more numbers between the interval [0,1] than in theset N.


We will stop at this second demonstration to give an idea of its significance.

Let us consider the set of numbers between 0 and 1. It also includes irrationalnumbers such as:

I1 ="2# 1 = 0.4142... I2 =

"3# 1 = 0.732... I3 = e# 2 = 0.71828...

The interval [0,1] also includes rational numbers such as 12 ,

13 , etc., but these

numbers can also be written in the form:

A = 0.abcd...

We can admit, in particular, that:

12 = 0.5000000000...

We then show that the set of numbers of the type “A” lying between 0 and 1 is non-countable. The demonstration can then be carried out by working through a reasoningby absurdity in the following manner: let us assume that we have a table, T , of allpossible numbers, A, each of which has an order number (see the following):

Table T

A1 = 0.a1 b1 c1 d1...........

A2 = 0.a2 b2 c2 d2............

A3 = 0.a3 b3 c3 d3.............

.............................................

An = 0.an bn cn dn............

...............................................

If we demonstrate that we can construct a number that we denote by:

X = 0.x1 x2 x3 x4.............

different from all numbers An of the table T , then we would have proven that T doesnot contain all the positive numbers from A, lying between 0 and 1.

This is where Cantor’s famous “method of the diagonal” appears.


By following the oblique line (diagonal), we can construct the number:

B = 0. a1 b2 c3 d4...

We still cannot say, however, that B is surely different from A1 (because the firstnumber a1 is the same in B and in A1), and we know nothing of the other numbers(which may, after all, be partly the same in B and in A1). But if we replace a1 by anumber x1, different from a1, this gives:

B1 = 0. x1 b2 c3 d4...

thus, we are certain that B1 is different from A1. In order to now have a numberdifferent from A2, it is enough to repeat the method and replace, in B1, the number b2by a different number x2. We then have:

B2 = 0. x1 x2 c3 d4...

and this number will also be different from A1 and from A2.

Figure 9.1. The “method of the diagonal”

It is evident that we can continue this construction indefinitely and finally obtain anumber X , such that X is different from A1, A2, A3, etc. (see Figure 9.1).

It results from this that as X is different from all numbers contained in the tableT, X is not in the table T . The table T , which contains a countable infinity of numbers,does not then contain all the numbers lying between 0 and 1. The set of real numberslying between 0 and 1 does not, thus, have the power of the countable. It has a higherpower, which we will denote by .1. The idea is that the series of real numbers lying


between 0 and 1 “exhausts” the series of whole numbers. We have here an infinity ofhigher order.

As we said earlier, Cantor demonstrated that the interval [0,1] has the same poweras the set of real numbers (the real line, in the geometric plane). And we can alsoshow, in a more general manner, that the set of points on a line has the same power asthe set of points in the plane, in a volume, etc., such that the set of real numbers, R,has the same power as Rn. This is the power said to be the continuum.

9.2.3. Continuum hypothesis and generalized continuum hypothesis

A final question may be briefly raised: what is the relation between .0 and .1,and, more generally, if there exist powers superior to (.2,.3, ...,.n), between .n and.n+1?

To respond to the first part of the question: we can consider that a real number isa part of the whole, an infinite fraction that is expressed in the form of an unlimiteddecimal, either periodic – and it is rational – or non-periodic, and it then correspondseither to an irrational or to a transcendental. In all cases, a real is indeed a part of thewhole. Cantor’s idea then was to posit that the set of real numbers was the set of partsof the set of whole numbers. We demonstrate, moreover, that if a set has n elements,the set of its parts possesses 2n elements. Cantor then legitimately assumed that if .0

was the power of the set N, then the power, .1, of the set R must be such that:

.1 = 2)0

This is called the continuum hypothesis (abbreviated to CH). This hypothesis can,as is obvious, be generalized, as Hausdorff did in 1908, positing that:

.n+1 = 2)n

which is generalized continuum hypothesis (abbreviated to GCH).

Cantor genuinely tried to demonstrate these hypotheses. His failure to do thisplunged him into a particularly depressive period, where he renounced mathematicsand began to teach philosophy. We now know, however, that such a demonstrationwas not possible.

But we also know since Gödel (1940) that if the Zermelo–Fraenkel (ZF) axiomaticon set theory is consistent (non-contradictory), then ZF + GCH is also consistent.

And we have also known, since Cohen (1964), that ZF is consistent in the CH.


It follows from this that we may, or may not, depending on the case, take the CH(or GCH). In the case where we do not take it, the continuum may be any of the alephsand not necessarily .1

5.

Let us conclude by recalling that Cantor did not only explain the concept of actualinfinity in the domain of power or the cardinality of sets. He also constructed a theoryof infinite, ordinal numbers, with the associated operations that make it possible tocount using these numbers and, thus, manipulate infinite quantities.

It is, consequently, clear that it was after Cantor that actual infinity moved out oftheological books for the first time and acquired a status in its own right inmathematics.

9.3. The development of set theory

Via these considerations on the infinite, the set theory introduced intomathematics abstract and very general language, served by a range of new symbols,some of which were known and others which were more recent. Thus, theintersection or union, the existential quantifier or the sign of belonging had alreadybeen used by Peano, the universal quantifier was used by Russell in 1902, thebrackets surrounding the elements of a set were used by Zermelo in 1907, Russellused the symbols p, q, r to designate propositional variables in 1903 and, finally,different signs of negation and the principal connectors as we know them todayemerged between 1908 and 1910. On the other hand, the arrows used for “resultingin” or “if and only if” are more recent and owe their existence to Bourbaki in 1954,while the sign for an empty set (/) was proposed by Weil and only appeared in 1939.Various other symbols that are currently in use, such as the box indicating the end ofa proof (the “Halmos”, named after Paul R. Halmos) also date to the 1950s. Thealephs, of course, emerged thanks to Cantor and inclusion, a sign deriving from “C”was thanks to Schröder and has been used since 1816 by Gergonne to signify‘contains’ [GER 17, pp. 189–228].

The history of the set theory after Cantor is particularly bumpy. It was born out ofreflections on the concept of the infinite, which were also presented by Dedekindd[DED 71, DED 76a, DED 76b] and Riemann’s reflections on the concept of space[RIE 53, RIE 68a, RIE 68b], set theory, which as we have seen was constructed byCantor in the 1880s, sowed discord among mathematicians when they discoveredthat it led to paradoxes (set of all sets, sets of sets that do not contain themselves as

5 With, however, major consequences on the resulting mathematics that make certain choicesinconvenient and others less plausible.


elements, etc.)6. In the early 20th Century, the “crisis of fundamentals” revealed itsweaknesses (which were rapidly identified as arising due to the lack of restriction onthe use of the concept of set) and made it essential to axiomatize it. Once this wascarried out (Zermelo’s work [ZER 04, ZER 08]), the theory regained momentum anddeveloped into its classic form.

The period 1900–1930 thus saw the acceptance of the set theory as a new,completely indispensable field of modern mathematics. From the 1900s onward,stimulated by the problems arising from analysis, a great body of work was carriedout in this field by the French mathematicians Borel, Baire and Lebesgue, whosecontributions led to the development of what is today called the descriptive theory ofsets. Extending Cantor’s work on definable sets of real numbers (for which he hadestablished the validity of the CH in the case of closed sets), these authors aimed tofirst prove this hypothesis. It was at this time that Borel’s hierarchy of sets, Baire’shierarchy of functions and the concept of the “Lebesgue measure” (a crucial tool inmodern analysis) were introduced.

To define this current in a more precise way, we can say that the descriptive settheory is the study of certain types of definable sets of real numbers that are obtainedfrom simple types (such as open sets or closed sets) using well-known operationssuch as complementation or projection. Borel sets were the first hierarchy of suchsets. Introduced in 1898 in one of Emile Borel’s texts [BOR 98], they were obtainedfrom open sets using a repeated application of countable union and complementation.In 1905, Lebesgue had studied these “Borels” in a dissertation that made its mark[LEB 05, LEB 72]. He notably showed that their hierarchy possessed levels for allcountable ordinals and presented Baire’s functions as the perfect counterparts to theseBorel sets.

The main objective of the descriptive theory of sets was to find the structuralproperties common to all definable sets. For example, it was shown here that theBorel sets had the property called the “perfect set”7. In the case where they wereuncountable, they accepted a perfect subset and, consequently, in both cases, werefound to be compatible with the CH. This result was obtained in 1916 by bothHausdorff and Alexandroff, independent of each other. Other important “regularity

6 We will remain silent on this question that has seen much discussion among philosophers.7 In a topological space, a perfect set is a closed part without an isolated point, or, equivalently,a part equal to its derived set (the set of its limit points or accumulation points). Thus, theempty set and Cantos space are perfect sets. Generally speaking, any completely metrizable,non-empty perfect space contains a subspace that is homeomorphic to Cantor. We also knowthat any locally perfect, compact, nonempty space contains a subset that has the same power asCantor space. In both cases, the considered space has at least the power of the continuum.


properties” studied in this descriptive theory of sets were identified: for example theproperty of being “Lebesgue measurable”, as well as the property called “property ofBaire” (namely differing from an open by a meager set or a set of first-category).

At this period, one of the topics that drew a lot of interest was the study of“analytic” sets. These were the continuous images of Borel sets or, equivalently,projections of Borel’s projections. In 1905, the young Russian mathematicianMikhail Suslin discovered an error in Lebesgue’s dissertation, proving that theprojection of a Borel set is generally not a Borel set. However, he was able toestablish that the analytical sets also possessed the property of a “perfect set” andthus, verified the continuum hypotheses8.

In 1923, Nikolai Lusin and Waclaw Sierpinski studied sets called “co-analyticsets”, which led to a new hierarchy of projective sets, which began with analytic sets()

11), their co-analytic complements (ensembles $11), the projections of thesecomplements (

)12), the complements of these projections (set $12) and so on.

During the 1920s, many research projects were carried out into these new typesof sets, chiefly by Polish mathematicians around Sierpinski and the Russian school ofLusin and its students. An essential result that Sierpinski obtained was that each set)

12 is the union of .1 Borels (this is the same for the sets)

11). But this kind ofresearch, which had become traditional for this topic, would then stagnate until 1940.

Lusin, Sierpinski and their colleagues would soon encounter extreme difficultiesin their work. Highly discouraged, Lusin even concluded (in an article in 1925) thatwe did not know and would perhaps never know if projective sets had the desiredproperties of regularity [KAN 95].

These commentaries, which are revelatory, if seen through the prism of laterdevelopments, led to hypotheses that made it possible to resolve all these questions(see, for example, the axiom of projective determination9). They also highlight the

8 These results appeared in [SUS 17].9 Alternative to the axiom of choice, which leads to the existence of non-measurable sets,the axiom of determination (AD), introduced by Jan Mycielski and Hugo Steinhaus in 1962,stipulated that for games of perfect information, of length $, played by two players in atopological space (topological games) – a situation in duality with the questions posed byBaire – each game of a certain type was determined, which meant that one of the two players hada winning strategy. AD implied that all the subsets of real numbers are Lebesgue measurable,possessing the property of Baire and that of the perfect set, implying a weak form of the CH.AD also implied the consistency of ZF, but it was not possible to prove the consistency ofZF+AD relative to ZF and the presence of AD necessarily implied the negation of the GCHas the GCH implied the axiom of choice. In the late 1980s, AD would be replaced by PD (the


difficult methodological and philosophical questions that these recent hypothesesbring up, namely the problem related to the kind of proofs that support thehypotheses.

Lusin summarized the state of the art in a book that came out in 1930 [LUS 30]and would become a reference in the years to come. Since this work came out it hasbecome quite common to present the results of the descriptive theory of sets for theBaire space ($ of infinite sequences of natural numbers, which were in fact,established by René Baire in his thesis of 1899 [BAI 99] (a result that had beenanticipated by Dedekind around 1891). The Baire space is provided with a certaintopology that makes it homeomorphic to the set of irrational numbers, and expertsconsider it to be perhaps the most fundamental object in the study of set theories,alongside the set of natural numbers [MOS 94].

This current of the descriptive theory of sets must be counted among the mostimportant contributions made by set theory to analysis and topology. But it would bediscovered that what had begun as an attempt to prove the CH could not, in reality,achieve this objective. It would soon be demonstrated, using the axiom of choice, thatthere are sets of non-Lebesgue measurable real numbers [VIT 05a], as well as sets ofuncountable real numbers without any perfect subset [BER 08]. Results of this kindclearly proved the impossibility of achieving the stated objective by focusing on setsof definable reals.

Additionally, thanks to Gödel’s work in the 1940s (and also thanks to the“forcing” technique introduced by Cohen in the 1960s) we can understand whyresearch in the 1920s and 1930s had stagnated: the new fundamental results ofindependence showed that theorems established by Suslin (property of the perfect setfor analytic sets), Sierpinski (sets

)12 such as the unions of .1 Borel sets) as well as

certain others were the best possible results that we could obtain based on the ZFCsystem of axioms. Such a result is philosophically significant: an exploration of theworld of definable sets, starting from open (or closed) sets through complementation,countable union and projection, was enough to attain the limits of the ZFC system.

axiom of projective determination). This stipulated that for an infinite game with two players,with perfect information and of length $ and in which the players play with whole numbers, ifthe winning set (for either player, knowing that the project sets are closed for complementation)is itself projective, then one of the two players has a winning strategy. This axiom is not a ZFCtheorem but, unlike AD, it is not inconsistent with ZFC and implies that all the projective setsare Lebesgue measurable, have the property of Baire and that of the perfect set. It also impliesthat all projective binary relations may be “standardized” by a projective set (weaker form ofthe axiom of choice). The PD itself follows from certain axioms on large cardinals, such as theexistence of the infinitely many cardinals of Woodin.


From this we have the necessity of new axioms to go further, a necessity that Gödelhimself had emphasized after the Second World War.

Another important legacy Cantor bequeathed us is the study of transfinitenumbers, certain elements of which were presented earlier. In 1908, Hausdorff, whoworked on the types of uncountable orders and introduced the GCH (see earlier) alsoenvisaged the possibility of an “exorbitant” cardinal that today would only be called“weakly inaccessible”, that is a regular cardinal, but one that is not a successor (acardinal + is called “regular” if the decomposition of + into a sum of small cardinalsrequires +-many of these numbers). Some time later, in the second decade of the1900s, Paul Mahlo, who studied hierarchies of large cardinals, pioneered a new fieldthat would become a central zone for set theory. He obtained a succession ofinaccessible cardinals using a certain operation that implied the concept of“stationary subset”. These cardinals have, since then, been called “Mahlo cardinals”.However, the study of these cardinals developed very slowly. During this time,Hausdroff’s textbook titled Grundzüge der Hausdorff Mengenlehre (dating back to1914, reprinted in 2002) introduced two generations of mathematicians to set theoryand general topology.

The greatest advances, in this “very large infinity”, discovered by Hausdorff andMahlo, would not take place before 1930. The concept of “strongly inaccessiblecardinal” would then be defined by Sierpinski and Tarski, then by Zermelo. Astrongly inaccessible cardinal is a regular cardinal + such that 2x is lower than +each time that x < +. Although the weakly inaccessible cardinals involved onlysimple closing for the “successor” operation, strongly inaccessible cardinals imposedthe concept of closing much more strongly for the “powerset” operation. In the sameyear, in an avant-garde document on the ZFC models, Zermelo established the linkbetween non-countable highly inaccessible cardinals and certain “natural” ZFCmodels. He assumed, notably, that the “powerset” operation is fully determined.

In the same year, Stanislaw Ulam followed considerations from analysis (measuretheory) to arrive at a concept that would also become central to set theory: that ofthe “measurable cardinal”. It was seen that these cardinals, defined by a property ofthe measure theory, must also be highly inaccessible. In effect, many years later, thiswould be established (by Hanf, working on earlier work produced by Tarski) that thefirst inaccessible cardinal was not measurable, which showed that these new cardinalswere even more “exorbitant” than had been believed.

As we can see, the Polish school led by Sierpinski [SIE 76] thus played a majorrole in the development of set theory in the interwar period. Measurable cardinalsbecame particularly important toward the end of the 1960s, when it became clear thatthe existence of a measurable cardinal, in fact, contradicted Gödel’s axiom ofconstructibility V = L. This justified Gödel’s condemnation of the new axioms,expressed in what is sometimes called “Gödel’s program”.


Set mathematics would continue to develop in the axiomatic and powerfulstructural approach, which would dominate a major part of the 20th Century. Arecent textbook on set theory brought together the fields that we have just discussed,augmented by several more specific subjects such as combinatorial set theory,Gödel’s theory of constructible sets, large cardinals and their “internal” models,stationary sets, giving the slight impression of a “mathematical supermarket” where,depending on the case, we could take or leave such and such an axiom from therange. And though there often exist in these books historical notes summarized at theend of the chapters, they do not make it very easy to understand the chaining of ideas[JEC 00].

9.4. The epistemological route and others

In this context, while in the first half of the 20th Century, at least one philosopher(Jean Cavaillès) took an epistemological view of the emerging set theory [CAV 38a,CAV 38b, CAV 62], very few philosophers were sagacious enough to understand whatthis theory involved and to use it for philosophical ends.

Cantor was probably the last mathematician to have made the link betweenmathematical infinity and theological infinity. Nonetheless, as we have seen, it wasbecause of pressure from the Church in the form of Pope Leo XIII, who insisted onplacing God above the mathematician, that he distinguished between “uncreated orabsolute eternal infinity”, which could only be attributed to the Supreme Being(infinitum aeternum increatum sive absolutum) and the infinity (or transfinity) createdby the mathematician (infinitum creatum sive transfinitum), a simple humanconstruction [THU 77]. However, what the plurality of infinities shows, just as withthe paradoxes related to the “set of all sets” expressions, “set of all alephs” or “set ofall ordinals”, was indeed the perfectly rational rejection of an entity that includedeverything. And if, of course, the God of modern theologists cannot be compared to aset [TAP 11], the fact remained that doctrines such as panentheism, and maybepantheism, which were formerly jointly supported by a philosopher such as Spinoza,were rendered impossible by set theory, which turns them into paradoxes incarnate.And as for Tapp’s suggestion of a sort of symbolic parallel between the fact that wecan no more embrace God than we can the ordinal class [TAP 11, pp. 13–14], it isboth granting much and risking much, as we gain absolutely nothing from aphilosophical point of view or from a theological point of view by comparing God toinconsistant multiplicity.

In any case, it does not seem as if the concept of infinity was a major preoccupationfor philosophers in the second half of the 20th Century. Like Heidegger, they preferredemphasizing human finitude (see, in particular, Paul Ricoeur’s early works [RIC 60]).


Only Jean-Paul Sartre’s “practical set theory”10 seemed to explicitly refer to set theorywithout, moreover, preserving or transposing anything from the abstract set theory.Sartre’s words, on the contrary, clearly and carefully distinguished “practical” sets,that is taken in the dialectic of becoming, from “abstract” sets defined once and for all.In life, “groups” and even “couples” are made and unmade, “queues” form and meltaway, “classes” confront each other and the property of culture – and maybe politicsin general – is to allow individuals to escape from the determinations that fence theminto “sets” that they wish to escape precisely because, being from the living world,they cannot exactly be part of the “pratico-inert”. But we clearly see that the conceptof “set” is not the most appropriate one for posing the problems that Sartre reflectedon, just as it did not resolve the questions posed by Levinas, whose reflections, on theother hand, were focused on escaping from the concept of totality11. Indeed, it was theconcept of an “evolving dynamic system” that allowed him to take into account these“detotalized totalities” that interested him. Moreover, questions related to the axiom ofchoice, which were only problematic for “infinite sets”, do not, in principle, concerndaily life. The choices we need to make in everyday situations are generally finitein number12. Similarly, in daily life we cannot fear the negative consequences of thisaxiom in mathematics (the existence of non-measurable sets). We can, at best, observethat the choice of the AD in a model of daily life could bring about the possibility offinite or infinite games in the sense of the formulation used in a book by James P.Carse [CAR 88]. But the “infinite player”, who plays to continue the game and notto win, has little in common with the protagonists of the “topological games” of thedescriptive theory of sets. It must be recognized that recent philosophers have under-utilized or even completely ignored the subtleties of the abstract set theory.

This, however, was not the case with Husserl, who was almost a contemporaryof Cantor and of Riemann. The Mathesis Universalis that he aimed to construct, andwhich included, in particular, a formal doctrine of science, was based on mathematics,notably that which was later called the theory of multiplicities.

10 Subtitle of the Critique of Dialectical Reason [SAR 60]. For more on this subject, see ourcommentary in [PAR 04].11 See, for example, [LEV 61]. We have shown in another book that the postulate of the author,according to which we can not put in the same set two entities as different as another person andI (which, for Levinas, can not be added) amounts to questioning the axiom of arithmetic. Thisis the same as implicitly using the idea of “inconsistent multiplicity”, a concept with which it isdifficult to work with in mathematics. For another possible formalization, see [PAR 15b].12 Let us note, however, that the model of Platonic dichotomic processes, which requires thatwe be able to define the set of possible “filters” over a set, that is the “ultrafilter” that containsthem, assumes the axiom of choice in reality (see [PAR 86]). We may imagine that havingcomplete information on all possible choices that can be envisaged, in a given situation and at agiven point of time, could also involve the axiom of choice.


The concepts of “multiplicity” (Mannigfaltigkeit) and of “the theory ofmultiplicities” (Mannigfaltigkeitslehre) first appeared in the author’s work toward theend of the first volume of his Logical Investigations. “Mannigfaltigkeit” designates,here, “a possible field of knowledge in general” [HUS 69, p. 274], governed by aformal theory. The concept was from Riemann, who justified its use in his famousdissertation at the Royal Society of Sciences in Göttingen, titled “On the Hypotheseswhich Underlie Geometry” [RIE 68c, pp. 280–299]. It shows, notably, how essentialthey are to several recent mathematical fields (the study of analytical functions withmany variables, Abel’s theorem, the work of Lagrange, Pfaff and Jacobi, the generaltheory of differential equations, etc.) and even introduced the concept of“n-dimensional multiplicity”.

At this time, a period after Formal Logic and Transcendental Logic, Husserlemphatically differentiated this from the concepts of “Inbegriff” and “Menge”, whichwould become almost synonymous with “Kollektion”. But going beyondinvestigations into the transcendental structures of subjectivity, which are developedin this book, the goal kept in mind was establishing a formal doctrine of science,founded on the study of multiplicity.

This goal would even be retained much later, if we read attentively, in a lecturethat Husserl delivered at the Kulturbund in Vienna on May 7, 1935. This was on“The Crisis of European Humanity and Philosophy”. This famous text elaborated onphilosophy and its ramifications (the different specific sciences) as a special class ofcultural creations (Kulturgebilde), involving the idea of an “infinite task”. This wasthen further divided into an “infinity of tasks” for which Greek geometry had, overhundreds of years, been able to provide the example and golden model. Indeed, datingfrom antiquity, mathematics was revealed to be a task full of meaning, opening ontoinfinity and which had as correlate “new man with infinite goals”.

As Husserl saw it, Greece as a political community was also the anticipation ofthe supranational totality that, according to the philosopher, Europe could become,with its hierarchy of social structures obeying the infinitist spirit. This was, however,provided people agreed, instead of tearing each others’ theories apart.

All this happened, therefore, as though the paradigm that implicitly directedHusserlian text was, here, G. Cantor’s theory of infinite sets, which was itself theprototype for a theory of multiplicities. This theory, through the GCH, manifested notonly the existence of an infinity of infinites of different powers, but also their possiblearticulation of one another (.n+1 = 2)n ). The idea of a process of generation ofsuccessive infinities – which we find in the text of the Krisis – like that of anindefinite hierarchy of different infinities, included in a collection that is not a set but,as Cantor wrote to Dedekind, an “inconsistent multiplicity”, visibly finds its echoesin the German mathematician’s reflections.


In the second part of the lecture, Husserl would even postulate that philosophy, inits consciousness of the supreme self, must in fact think of itself as a “branch” of theinfinite task (einem der Zweige der unendlichen Aufgabe) in which humanity hasbeen involved from the time of the Greeks. And even if no philosopher can thinkthemselves capable of becoming master of the true and complete meaning ofphilosophy, of embracing its infinite horizons in all their “ampleur”, nonetheless thephilosophical task, a branch of the infinite task, must be laid out within it as in aninfinite tree13.

For Husserl, the outcome of the crisis into which infinitude plunges us is dubious.Assuming that we can overcome the naturalism and objectivism brought about by thescientific step, what remains would be to elaborate on the concept of Europe and bringin the historical teleology that must order the infinite goals of reason. But how can weensure the possibility of such an ordering of these goals? As we know, the idea that anyset may be ordered, which is expressed in Zorn’s lemma on set theory, is a statementthat is strictly equivalent to the axiom of choice. It is thus a simple possibility that weare well within our rights to refuse.

It is not easy to know whether Husserl perceived all the consequences of theimplicit model that he used. A priori, we do not find any explicit allusion in his texteither to the Beth theorem or to infinite trees in general (in spite of the fact that at thistime, a theory of infinite ramified ordered sets had already been developed by theRomanian mathematician G. Kurepa [KUR 35] and that many different infinitetrees – notably that of Aronzsajn – were known). Similarly, the literature surroundingthe “Suslin problem” (are all contained ordered sets identical from the point of viewof the order, to the mathematical continuum?) seems to have passed him by. MikhailSuslin, a student of M. Lusin, had generalized Borelian sets to invent analytical setswhose theory cast some light on the mathematical continuum and would have helpedilluminate Husserl’s crucial question of the order of infinite tasks14. But Husserl was,above all, obsessed with the risk of renunciation and the outbreak of violence thatwould sweep across Europe and bring in war that would stay15.

13 We can even add that its own infinity can be easily deduced because, when a tree is infinite,a known theorem, proven by the logician E. Beth, states that one of the branches, at least, isinfinite [BET 72, pp. 80–81].14 For more information on this question today, see [SHE 90] and [JUD 92].15 For more information on all this, we take the liberty of referring to our more detailed article[PAR 94].


9.5. Analytical philosophy and its masters

The logical reflection that accompanied the question of the foundation ofmathematics sparked off a large movement in philosophy, especially in the Germanicand Anglo-saxon countries. The scope of this movement widened internationallyover time, especially through the English language, which spread far and wide. Aswe cannot give the reader a complete overview of analytical philosophy, or even asuccinct summary of the field, let us just recall a few of its main propositions here.

Founded on the new, contemporary logic from the works of Gottlob Frege andBertrand Russell16 in the late-19th and early 20th Centuries, analytical philosophyhad the aim of clarifying large philosophical questions through a logical analysis oflanguage that was assumed to highlight the reasoning errors of philosophers whogenerally based themselves too naively on language.

The principles of analytical philosophy were clearly stated by Wittgenstein inTractatus Logico-philosophicus [WIT 01]: the aim was, essentially, to develop acritique of the philosophical use of language, which he believed very often usedterms bereft of meaning. That is, variables to which, through a lack of vigilance, wehad failed to attribute “value”. “To be”, Quine once said [QUI 53], is nothing but “tobe the value of a related variable”. This reduces existence to a logical existence –something Kant had, however, denounced as a paralogism that was at the root of theontological pseudo-proof for the existence of God.

The reduction of metaphysics to logic – a project that we could call “logicalpositivism” in its strongest sense – was a method that was particularly followed bythe members of the Vienna circle, who strove to draw a boundary between rigorousexpressions (that were, thus, “scientific” as they were stated in well-formed formulasof a logical language) and others (meaningless). Thus, Frege said that a propositionsuch as “the present King of France is bald” was meaningless as the referent did notexist.

In the earliest analytical philosophy, natural languages were seen as any otherartificial languages, depending on the rules for the logical first-order predicatecalculus. From this point of view, the two principal forms of logical reasoning – thesubstitution principle and the rules of inference – had to be rigorously applied tonatural languages to get rid of confusion or approximations. Thus, the majordistinction that G. Frege made, in his writings on philosophy and logic, was between“meaning” (Sinn) and denotation or reference (Bedeutung). This made it possible todenounce certain expressions: statements such as “evening star” and “morning star”

16 See [RUS 10]. And for a presentation of Russell’s theory up to the Principia, see [PAR 12a].


certainly did not have the same meaning, but they both denoted the same referent (theplanet Venus) [FRE 71].

In this context, the English philosopher, Bertrand Russell [RUS 05] explored,among other things, how to form good definitions. For example, defining the conceptof number, which must evidently not use any reference to this concept itself in thedefinition, to avoid all petition of principle.

A number of problems, however, soon arose. Thus, the substitution principle wasfound to be flawed for statements that contained what are called “propositional attitudeverbs” (believe that, fear that, etc.). The statement, “Galileo believed that the planetaryorbits were circular” is already one that defeats the substitution principle. Indeed, asthe planetary orbits are elliptical, substituting “ellipse” for the term “planetary orbits”produces an absolutely false statement: “Galileo believed that ellipses were circular”.Similarly, the elimination of proper nouns also posed a number of problems. Thus,the sentence “Scott is the author of Waverley” was interpreted by Russell to mean,“There exists only one x and x is the author of Waverley”, a sentence that, in allrigor, leads to identifying Scott as the author of Waverley. This would logically leadto substitutions within itself that would make it completely tautological, producingsentences of the type “Scott is Scott”, or “The author of Waverley is the author ofWaverley”. Moreover, it is clear that a nearby statement such as “I saw Scott” cannotbe replaced by “I saw the author of Waverley”, as the person who utters this statementdoes not necessarily know whether the Scott they are talking about is the author ofWaverley.

These difficulties, and others of a similar nature, which sparked off endlessdebates within the analytical community, generated a large body of work whosecontent was, however, quite slim as it spoke about statements that were, in general,completely devoid of philosophical interest, or even devoid of any interest. The strictlogico-linguistic parallelism defended in the early analytic philosophy was thusrevealed to be more and more fragile, to the point that the second Wittgensteincompletely dissociated himself from the arguments laid out in the Tractatus,according to which logic was considered as the mirror-image of the world.

Following these critiques, analytic philosophy evolved more toward a study ofordinary language. Natural language, which was considered a “poor tool”, simplistic,error-riddled and needing to be reformulated into formal, rigorous language that wasfree of ambiguities, soon proved itself to be much more complex than it had initiallyseemed. Despite attempts by Carnap [CAR 02], and then Quine, to extend the use oflogic to the syntax of natural languages, this path was gradually abandoned.

Although certain distinctions in analytical philosophy remained perfectlyacceptable (the different uses of the word “to be”, for example, sometimespredicative, sometimes affirming existence, sometimes revealing a simple identity),


the general attempt to formalize statements in natural language was completelyabandoned in its strongest form. The discovery of certain specificities of language(e.g. the illocutionary nature of certain statements, such as the language used to“make certain things be”: I name this boat the “Joseph Stalin”) even promoted thedevelopment of metaphysics, rather than rejecting it or going beyond it. On review,therefore, analytical philosophy did not gain much eminence beyond countries withan Anglo-Saxon tradition, where traditional philosophy, especially that of Hume, wasassociated with empirical and already anti-metaphysical traditions to contest theambitions of continental tradition.

We can even propose that the project of the “scientific” philosophy thus created,that is one whose methods and exposition are liable to be held to the same standardsas scientific discourse, is a double failure. Not only do any number of everydayphenomena (e.g. the entire field of literature and, more generally, aesthetics) falloutside the scope, at least partially, of the drastic criteria for analysis, but theanalysis, immovably stuck in its own internal discussions, was never able to developinto an authentic philosophy and finally degenerated into a simple reflection onlanguage and its uses.

The attempt to reduce language to logic, which was in itself based on an error (thepossibility of a perfect reduction of mathematics to logic), was found to contradict notonly facts, but also theory, as Gödel’s work demonstrated the irreducibility of one tothe other.

The analytic tradition, however, passed on a style of writing and approachingphilosophical problems that were based on language and that gradually spread to allschools of philosophical thought, from the philosophy of science to the philosophy ofethics and even theological thought. The initial project, that of eradicating themetaphysical to promote science, then turned into its opposite – the interminablequibbles of Anglo-Saxon philosophers, leading to new forms of scholarship andfinally reinforcing the hermeneutic tradition.

Moving slowly away from science and its structures, albeit with no true knowledgealternative being created, analytical philosophy saw its pertinence decrease at the sametime that logic itself, far from limiting itself to standard predicates, had begun to divideinto a variety of rival models, finally abandoning any normative goal.

The “model-theoric” interpretation of modern logic, just as with its increasinglyobvious relation to the computer sciences, would be the final blow. And just as thecreative power of mathematics broke free of the increasingly narrow bounds of logic,natural languages also proved to go beyond any kind of subservience to formalisms,which (to say the least) were in themselves problematic to interpret.

And in a situation where the flexibility of natural languages and their expressivecapacity made it possible to gather information from global human experience that


the analytic tradition, understood stricto sensu, could not abandon, traditionalphilosophy had no trouble working on a new lease of life, despite the fact that itsdeath had been often predicted. Forced to either abandon its project or rectify it,continental philosophy remained a possible alternative. This was of course subject tothe condition that it be informed by advances made in knowledge that, traditionally,had always sustained it. Independent of this drastic project, many major philosophersthus pursued humanistic research for a truth that made sense to a being engaged inthe world and that, beyond their more or less rigorous linguistic translation, mustconfront real problems.

9.6. Husserl with Gödel?

It remained that the form philosophy held must change and that its objective mustbecome more modest. The point that Husserl’s project and others of the same typestumbled against (we recalled earlier that the resurgence of the idea of Mathesis atleast as it was understood since Descartes, was seen almost at the same time in theworks of Peano, Frege, Russell and others) is, of course, Gödel’s work, with his resultson incompleteness in 193117 – a body of work that has, for the most part, been muchannotated since18.

From this point of view, it was quite significant to see the Husserlian project gainmore ground with respect to the Hilbertian project while, in parallel, from the 1930sonward, this was refuted in K. Gödel’s famous article, “on the formally undecidablepropositions of the Principia Mathematica and related systems” [WAN 87, p. 122].He then grew interested in Husserl’s work dating after 1907 but does not seem, at thisdate, to agree with the ideas of the philosopher. According to Hao Wang, Gödel firstthought that it was possible to progress in philosophy in the same way as infundamental science19, and then, changing his opinion, adhered to the methodrecommended by Husserl. However, “he probably did not accept the emphasis thatHusserl placed on subjectivity and wanted to use the method to arrive at ametaphysical system that would be similar in scope to Leibniz’s, Monadology, but

17 Gödel’s work has now been published: see [GOD 95].18 Apart from the now-classic work by J. Ladrière [LAD 57] we can also refer to [NAG 89].19 Perhaps, we should associate to this specific period his initial, infamous idea of the “proof”of God’s existence, founded on the S5 modal system of logic. An initial version was found in anotebook dating to 1941, but this was communicated to Dana Scott only in 1970. From this yearonward, the idea slowly spread everywhere without, of course, being greatly convincing. See[GÖD 95, “Ontological Proof”, pp. 403–404] and F. Nef’s [NEF 02] commentaries. We willnot comment here on this “proof”, which is discussed in the appendix. But it does not, strictlyspeaking, follow from Gödel’s mathematical work and is based on axioms that we can certainlychange in order to prove the inverse.


with more solid fundamentals...” [WAN 87]. It would, of course, be interesting toknow whether, in the period after 1931, Husserl got wind of Gödel’s work andwhether, if so, he drew any conclusions from this for his own philosophy.

Whatever it may be, this work did not prevent the development of mathematicaltheories other than the set theory, and even this made it possible to develop a structuralmathematics that only needed to express itself in all its power. We will now discussthis.

9.7. Appendix: Gödel’s ontological proof

The “proof” that Gödel proposed for the existence of God can be expressed intoday’s language in the following manner:

DEFINITION.– We say that x is divine (denoted by G(x)) if and only if x contains, asessential properties, all the positive properties and only these.

DEFINITION.– A is an essence of x (denoted by ess x) if and only if, for each propertyB, if x contains B, then A necessarily leads to B.

DEFINITION.– x necessarily exists if and only if each essence of x is necessarilyexemplified.

AXIOM.– Every property resulting from – that is uniquely implied by – a positiveproperty is positive.

AXIOM.– A property is positive if and only if its negation is not positive.

AXIOM.– The property of being divine is positive.

AXIOM.– If a property is positive, then it is necessarily positive.

AXIOM.– The necessary existence is positive.

From these axioms and those of modal logic, we then successively deduce thefollowing:

THEOREM.– If a property is positive, then it is possibly exemplified.

THEOREM.– The property of being divine is possibly exemplified.

THEOREM.– If x is divine, then the property of being divine is an essence of x.

THEOREM.– The property of being divine is necessarily exemplified.

In the formalization that follows, which combines these axioms, theorems anddefinitions, let us recall that " A signifies “A is possible” and ! A signifies “A is


necessary”. Moreover, in the references for the definitions, axioms and theorems, weomit the numbers for this present chapter (9) and the paragraph (6), such that axiom9.6.1, for example, simply becomes 1 (denoted by Ax. 1).

Ax. 1. P (,) 0! 1x[,(x) + -(x)] + P (-)Ax. 2. P (¬,) %&¬ P (,)Th. 1. P (,) + " 2x [,(x)]Df. 1. G(x) %& 1,[P (,) + ,(x)]Ax. 3. P (G)Th. 2. " 2x G(x)Df. 2. , ess x %& ,(x) 0 1-{-(x) + ! 1y[,(y) + -(y)]}Ax. 4. P (,) + ! P (,)Th. 3. G(x) + G ess xDf. 3. E(x) %& 1,[, ess x + ! 2x ,(x)]Ax. 5. P (E)Th. 4. ! 2x G(x)

Mechanized today, the proof is perfect. But it is, as is evident, relative tohypotheses that can be changed at will. We conclude this chapter, and this section,with this Gödelian “curiosity”.

PART 4

The Advent of Mathematician-Philosophers



Modern mathematics, which overcame several degrees of abstraction, sometimesgives the impression of being detached from reality. There are several reasons for this:

1) It does not seem to make up a body of truth relative to nature. The first failure,in this respect, appeared with non-Euclidean geometries. In 1827, Gauss was theperson to affirm that non-Euclidean geometries were applicable and that,consequently, they could rival Euclidean geometry. It thus followed that Euclideangeometry was but a possible representation of reality, one among others, such that“we can no longer be completely assured of the truth of Euclidean geometry” [KLI93, p. 161] nor of its aptitude to represent real space. We can also mention here thefact that Hamilton’s quaternions, with the non-commutativity of multiplication, thealgebra of Cayley and Grassmann, which were not necessarily associative, furtherintroduced strangeness into arithmetics. Similarly, Helmoltz showed theinapplicability of regular arithmetic to experience. For example, when we combine a100 Hz sound with a 200 Hz sound, we do not obtain a 300 Hz sound! And hence,we have this negative observation by M. Kline: “The sad conclusion thatmathematicians must draw from all this is there is no truth in mathematics, if by truthwe mean laws concerning the real world. The axioms on fundamental structures inarithmetic or geometry are suggested by experience. Consequently, these structureshave only limited application. Only experience can determine their domain ofapplication. The Greeks’ attempts, which consisted of trying to guaranteemathematical truths by starting from self-evident truths and only using deductiveproofs, proved to be in vain [KLI 93, p. 175].

2) Mathematics is not exactly the “pure diamond” that Plato, or Platonicmathematicians at any rate, used to imagine it was. As Morris Kline writes again, ifwe continue with the crystalline metaphor, mathematics is more “synthetic rock”[KLI 93, p. 178]. As the author shows, mathematicians must gradually concede thatthe axioms and theorems of mathematics were not truths for the entirety of physicalspace. Certain domains of experience suggest only sets of specific axioms that can be


locally applied, with their logical consequences, at any rate, in a sufficiently precisemanner so as to be used as useful descriptions. But if any domain expands, theapplicability of these axioms may become problematic. As far as the study of thephysical world is concerned, therefore, mathematics does not offer anything otherthan theories or models. And here, as is the case elsewhere, new theories can thenreplace old ones when experiment – or experience – shows that a new theory is closerto reality. The relationship that mathematics enjoys with the physical world is thenexactly what Einstein predicted in 1921: “For all that mathematical propositions arerelated to reality, they are not certain; and when they are certain, they do not relate toreality ... but it is, on the other hand, certain that mathematics in general andgeometry in particular owe their existence to our need to know something about thebehavior of real objects” [EIN 72, pp. 76–77]. The fact remains that vitaldevelopments such as non-Euclidean geometries and quaternions, which areseemingly discordant with nature, were finally found to be applicable. Why wouldhuman imagination, when totally freed of all restrictions, not find even morepowerful theories, if only because what succeeded once may succeed once again?

3) The vast expansion of mathematics does, of course, pose difficulties (evengreater for the philosopher than the mathematician) related to familiarizing oneselfwith multiple domains (including external domains such as physics), and the openquestions in science that many major mathematicians have worked on recently areoften difficult in theory. One temptation that young mathematicians face, then, is totake refuge in pure mathematics, all the more so that pure mathematics makes itpossible to find complete solutions to certain simple problems. The pressure frominstitutions and rules for research that have individuals competing against each otheroften result in professors suggesting that their students tackle problems from puremathematics that are unresolved but could be solved, and thus, we have a growingabstraction (multiple algebras are invented that may or may not find applications) andan intense generalization (curves of nth degree, spaces of infinite dimension, etc.).Specialization, which is another consequence of the institutionalization of thediscipline, may also bring in a certain sterility. Having said this, however, today’smathematicians are often tomorrow’s physicists, though this cannot be predicted. Forinstance, who could have said that Grothendieck’s K-theory would find application insuperstring theory? It is, therefore, absolutely essential, and even economical, todevelop mathematics for its own sake, independent of applications. But theepistemologist and even the philosopher must always reconnect those mathematicswith the real world, or even the living world (Lebenswelt in German).

This is why, in this section, we will first recall the basic concepts (which todaymake up area of “null Bourbaki density” – as J. Dieudonné called it) and then onlymention some of the aspects of the state of current research, especially in France,before suggesting how philosophy – and perhaps also mathematics, when it becomesphilosophical – could use these results.

10

The Rise of Algebra

Algebra in the modern sense, that is the study of algebraic structures independentof their concrete realizations, only emerged very gradually over the 19th Century, inconjunction with the general movement to axiomatize all of mathematics and theincreasing concern among mathematicians, since Galois, to substitute ideas1 withcalculations.

Up until this point, the essential purpose of algebra had been to resolve, usingexplicit formulae, what were called algebraic equations, that is equalities or systemsof equalities that, in addition to numbers, also contain variables called “unknowns”and whose value we seek to determine. Variables may appear in these equations in asimple form (in which case they are said to be of degree 1) or multiplied by themselves(in which case they are said to have a degree greater than 1).

Of course, these mathematicians – starting, as we have seen, from Descartes –strove to find methods to resolve these equations, but they ran up against a wall fordegrees greater than 5.

It is exactly these fruitless efforts to resolve general equations of a degree greaterthan 5, as well as other problems related to number theory, that then led to theintroduction of novel mathematical entities.

These entities were analogous to each other in their use and, consequently, themathematicians felt the need to discover what could be common to all these situations.They were thus led to think that the “nature” of the mathematical objects studied were

1 “Mathematics is ideas, and more precisely, the analysis of relations between these ideas andtheir extensions”, as I. Stewart wrote even more recently [STE 89, pp. 6–7].



essentially secondary and the English mathematician George Boole even declared in1847, in his book The Mathematical Analysis of Logic, that “mathematics processesoperations considered in themselves, independent of the diverse subjects to which theymay be applied”.

All through the 19th Century the process of axiomatization of algebra wouldcontinue, resulting in the structure we have today. This took place through variousstages.

10.1. Boolean algebra and its consequences

From 1850, English mathematicians very clearly defined the concept of the “lawof composition”2.

What would one day be called “Boolean algebra” possesses all the laws ofordinary algebra, but instead of applying to real sets, was defined over a domainrestricted to two elements, the set B = {0, 1}, equipped with addition andmultiplication. These operations are associative and commutative, 0 is the neutralelement of addition, 1 is the neutral element of multiplication. The latter is, as in realsets, right-distributive with respect to addition. But it is different for ordinarymultiplication as it is idempotent. In other words, for every x, for every y and forevery z, we have:

x+ y = y + x, xy = yx z(x+ y) = zx+ zy xn = x

Knowing that these operations can only be defined over B = {0, 1}, byintroducing an additional operation, subtraction, represented by the sign (#), we canadditionally abridge expressions of the type x+ y = 1 to x = 1# y (complementaryexpression) and x = xn to x# xn = x(1# xn&1) = 0 (which leads to the derivationof the non-contradiction principle – or the generalized non-contradiction – ofidempotency of multiplication).

Boole then applied this “algebra” – which he was, in truth, unable to really call an“algebra” as the concept had yet to be invented – to varied mathematical situations andentities: vectors, matrices, propositions and concepts in logic, probability theories3.But he did not stop there.

2 Here again, the work of G. Boole must be mentioned, with, however, a notable nuance. In hiswork The laws of thought, he used “operation” (+, #) for that which we today call “law” and“law” (associativity, commutativity, etc.) for the properties of these operations.3 The application of the theory of orders and the concept of the “Boolean lattice” wouldcome later. They assumed, as is obvious, the development of the concept of “lattice”, which

The Rise of Algebra 235

Its creator was convinced that binary arithmetic (a Leibnizian invention) wasisomorphic with the Chinese system Yi-king, and its aim was to convert the Emperorof China to western Christianity, notably to the central idea of ex nihilo creationbecause, as Leibniz said, “in order to produce everything out of nothing, one [thing]is sufficient”.

A less questionable application of what, in Boole’s own words, had become averitable “mathematics of the human intellect” appears in The Laws of Thought [BOO58, p. 187]. Having remarked that Spinoza, in the first part of his Ethics, presented aseries of binary distinctions that could be formalized using his new organon, Boolethen posits Axiom 1, according to which all things exist, either as things that are inthemselves (x) or as things that are in another (x!). This could be easily translated intothe formula:

x+ x! = 1, that is x = 1# x!

Similarly, Axiom 2, which distinguished between “things conceived of bythemselves” and “things that have been conceived of by another”, can be summarizedin:

y + y! = 1, that is y = 1# y!

Definition 3, now, differentiates between substance “which is in itself and bywhich it is also conceived”, from mode, “which is in another and conceived of by thisother”. We thus have:

z + z! = 1, that is z = 1# z!

Definition 7 also states that the universe if formed of things that are free (f ) andthings that are necessary (f ’), such that we have again:

f + f ! = 1, that is f = 1# f !

Moreover, through Definition 1 and Axiom 7, we can deduce that according toSpinoza, the universe consists of things that are causes of themselves (e) and thingsthat are the causes of other things (e’). Hence:

e+ e! = 1, that is e = 1# e!

would only come in with Birkhoff [BIR 67], and, via Stone’s isomorphism, would lead tocorrespondence with topology [PON 79].


But Definition 3 establishes, in fact, a perfect identity between substances (z) andthe things conceived of by themselves (y). We thus have:

z = y

Axiom 4 establishes the identity of the cause (e) and that by which something isconceived (y). Hence:

y = e

And Definition 7 establishes an identity between free things (f ) and those that existby themselves (e):

f = e

Definition 5 now states the identity between mode (z’) and that which is in anotherthing (x’). Hence:

z! = x!

As x = 1 - x’, the result of substituting z’ for x’ in this equation is:

x = 1# z!

from which we deduce:

x = z

In other words, the things that are in themselves (x) are substances (z). All theseresults can then be brought together into one single equation:

x = y = z = f = e = 1# x! = 1# y! = 1# f ! = 1# z! = 1# e!

One particular deduction from this is that:

z = 1# e!

which is none other than the expression for Proposition 6 from Book I of Ethics (asubstance cannot produce another substance). And, similarly, we deduce that:

z = e


which expresses Proposition 7 (it is in the nature of a substance to exist), a propositionthat Spinoza demonstrated in another way (through the identity of the substance andcausa sui).

Boole again shows that other propositions of the same order could follow thisformalization, whose expression can be seen in Ethics. Whatever we may think of thisattempt at rewriting, which certainly does not conform to the letter of the system, itcannot be denied that the advance made by Boole’s method did indeed have an impacton philosophy. We can again show this through the mathematician’s analysis of certaintexts by Clarke (one of Leibniz’s English correspondents).

A distant consequence of the existence of Boolean algebra would be P. DominiqueDubarle’s attempt, in the 20th Century, to formalize Hegelian thought based on theBoolean product ring {0, 1}2. This author makes this correspond to the four constantsof structure: U for universal, P for particular, S for singular, to which is added thenull term 0. Composed in this way, the set {U,P,0, S}n, equipped with the laws ofinternal composition, 3 (the transposition of the operation !) and !, with A!B =(A3B!)4(A!3B) (the transposition of the operation +), is that which Dubarle called“an ultraBoolean ring”. The Hegelian “circle of circles” thus becomes a “ring of rings”[DUB 70]. Here again, while philosophy historians may have some reservations, theimpact of formalisms on the understanding of the operations of philosophical thoughtis incontestable.

10.2. The birth of general algebra

Despite what we have seen, it would still be 1910 before the mathematician ErnstSteinitz’s vast synthesis would provide the abstract explanation that marked thebeginning of modern algebra, strictly speaking [STE 10].

In the 19th Century, from 1830 onward, it was the study of groups that woulddominate concerns of the time. Although this was introduced by Cauchy, it was Galoiswho would truly highlight this and who showed how important it was in the theory ofequations. This concept would then go on to play an essential role in almost all fieldsof mathematics, and find applications notably in crystallography, chemistry, physicsand quantum mechanics.

The work that German mathematicians carried out on algebraic numbers would,moreover, be at the origin of the study of bodies and commutative rings. Theseconcepts would thus appear as essential tools to algebraic curves and surfaces,leading to abstract algebraic geometry. Geometric language was thus introduced intocommutative algebra.


Linear algebra would grow in importance (after suitable axiomatization) whenmathematicians were able to perceive the linear character of many situations and theimportance of the process of linearization in others.

And as, according to Hilbert’s conclusion to his lecture in 1900, “mathematics isan organism and the condition for its vital force is the indissoluble union of itsparts”4, algebra would slowly (and successfully) join analysis through thesimultaneous consideration, on the same set, of algebraic and topological structures;this thus made up a particularly productive branch of mathematics that is calledtopological algebra.

We can then consider that the four fundamental parts of algebra were created at theend of the 19th Century: group theory, commutative algebra, linear algebra and non-commutative algebra, and finally, which we can call “topological algebra” (topologicalgroups, topological vector spaces). This last is the association of an algebraic structurewith a topological structure, making it possible to axiomatically work on problems thatarise from functional analysis. All these disciplines essentially yield tools that are veryuseful in mathematics and in physics. We will, here, only discuss the importance ofthe most fundamental of them all: the concept of group.

10.3. Group theory

The Greeks were interested very early on in their geometry by the properties ofregularity and we know that the crowning glory of Euclid’s Elements was theconstruction of five regular polyhedra, which, in substance, was the same asdetermining finite groups of rotation in three-dimensional (3D) space.

The concept of group, however, was anticipated by Gauss (who almost saw theadditive group of whole numbers, modulo m), and was only explicitly discussed byGalois, in the work he carried out on the resolution of algebraic equations “by radicals”in the early 19th Century. Developing an idea of Lagrange, Ruffini and Cauchy hadconsidered groups of permutations of the roots of an algebraic equation that wouldleave certain functions of these roots invariable. It was by taking this idea further thatGalois obtained his decisive results on resolution using radicals.

These initial groups were, therefore, finite groups and it was in the form of thistheory of groups of permutations that the general theory of finite groups began to

4 Lecture delivered during the 2nd International Congress of Mathematicians in Paris, 1900.H. Bénis-Sinaceur [BEN 87, p. 25, in particular] indicates that this metaphor was also used byDedekind (Gesam. Math. Werke, III, p. 430).


develop (notably in the work of Mathieu and Jordan [JOR 70]) until around 1870.The origins of mathematical crystallography (around 1830) would bring out otherfinite groups, this time formed by rotations and symmetries around one fixed point. Atthis stage, the most detailed results on the finite group theory were those discoveredby Jordan and Sylow. Much more recently, in conjunction with the preoccupations ofarithmetic and algebraic geometry, the finite group theory would see a new resurgence.The most spectacular discoveries of recent years were, however, those relative to whatwe call the characters and linear representations of these groups. We must mentionhere the work carried out by Brauer, Chevalley, Feit-Thomson and Novikov.

The earliest study of groups containing an infinity of elements can again be tracedback to the mathematician Jordan. This concept would gain considerable importanceduring the second half of the 19th Century. At this time, and in conjunction with therenewal of geometric studies and axiomatic concerns, the concept of thetransformation group would see a significant rise with the systematic study ofinvariables in such a group. That is, the study of properties that are not modified bythe transformations of the group. Thus, in our usual 3D space, the angles anddistances do not change with a displacement, the angles and ratios of lengths remaininvariable through similitude, the notion of parallelism or the nature of a conicsection is invariable through a regular linear transformation of coordinates. FelixKlein, in his famous “Erlangen program” of 1872, would clarify a general principlethat we will recall here in its intuitive form as follows: the data of a space and atransformation group operating over this space are enough to define a geometry. Ageometry is the study of properties that remain invariable when we applytransformations on the group. Thus, metric geometry (respectively, affine;respectively, projective) is the study of the invariable properties by the orthogonalgroup (respectively, affine; respectively, projective) and this theory constitutes acommon language that encompasses both Euclidean geometries and thenon-Euclidean geometries constructed at this time. Some time later, in the early 20thCentury, the theory of relativity would focus attention on a geometry constructedfrom a Lorentz group in a particular pseudo-Euclidean space that we call theMinkowski space. This plays an essential role in relativist and quantum theories.Klein’s work would also highlight the concept of isomorphic groups: in 1877, Kleindiscovered that the group of permutations of the roots of a fifth-degree equation issubstantially identical to the group of transformations of a regular polyhedron, calledan icosahedron. Although this concept of “isomorphic group” was technically firstused by Galois and by Gauss for particular cases, it would appear in its general formonly at this time.

It was not until the end of the 19th Century that the structure of the group, as it isconceived today, would finally be defined in an intrinsic manner (no longer restrictingitself to the case where the elements of the group are permutations or transformations).

From this time, the concept of the group has spread to every region of modernmathematics. First of all, the multiform character of the idea of the group was


perceived, largely going much beyond the initial concept of the group defined on aset of elements (topological groups, algebraic groups, schemas in groups and, moregenerally, “objects in groups” from a category representing a representable functor ofthis category in the category of groups). We have, moreover, discovered surprisingrelations between very different types of groups (e.g. between Lie groups, algebraicgroups, “arithmetic” groups and finite groups). On the other hand, experience hasdemonstrated the extraordinary efficiency of the concept of a group in all parts ofmathematics once we are able to introduce this concept here: the homology andhomotopy groups in algebraic topology, the principal fiber spaces in differentialgeometry and differential topology are well-known examples of this. Another, evenmore remarkable, characteristic is the possibility of defining a group structure overthe set of classes of differential structures that are compatible with a given(topological) manifold. This trend has also conquered physics. Although seeking toexplain experimental symmetries observed in atomic phenomena, theoreticiansnaturally turned toward group theory with remarkable success (see E. Majorana’sfirst steps and G. Mackey’s later theorization), although it was quite enigmatic: Whydoes nature obey principles of symmetry? Does nature really do this or is it the wayour brain perceives the world?

The structure of a group, in any case, is one of the simplest algebraic structuresand incontestably the most important structure of modern mathematics. Itsuniversality, moreover, is not restricted to mathematics or physics: Jean Piaget, theSwiss psychologist, highlighted the essential role that this concept played incognitive processes and Henri Poincaré stated that the concept of “group” pre-existedin our mind as geometry could not have been constructed without this. It did,however, take almost a century for such a concept to be made clear in its abstractform.

Let us recall that axiomatically a group is a set equipped with an internalcomposition law by which the element x5y corresponds with every couple of elements(x, y) in the set, with the following conditions:

1) This law must be associative (i.e. we must have: (x 5 y) 5 z = x 5 (y 5 z).

2) It must consist of a special element e, called the neutral element, such thatx 5 e = e 5 x = x.

3) It must be such that every element has an inverse (i.e. for every x, there existsan element y such that x 5 y = y 5 x = e).

Moreover, a group is said to be abelian, or commutative, if x 5 y = y 5 x. The name“abelian” comes from the name of the Norwegian mathematician Niels Henrik Abel(1802–1829), who had observed this possibility.

Let us give a few examples of this: the usual sets of numbers (relative integers,rational numbers, complex numbers) are abelian groups for addition; the sets of


non-null rational numbers or non-null real numbers are abelian groups formultiplication. An important example for a non-commutative group is that of thetransformation group for our usual 3D space that preserves the distance between twopoints: as we have seen, these transformations are what we call displacements. Theymake up a non-abelian group if we agree that the product of two transformations Sand T , S 5 T , is the transformation obtained by successively carrying out thetransformatin T and then S.

The study of classical groups has also led to the theory of algebraic groups,which admits important applications in algebraic geometry and in modern numbertheory. Through the action of Sophus Lie and his students, and then the work of theFrench mathematician Elie Cartan, the theory of transformation groups or“continuum groups” would plant the seed of one of the theories that is central tocontemporary mathematics, the Lie groups theory5.

To sum up, group theory is a theory that is central to not only contemporarymathematics, but all disciplines where mathematics is applied and where we candiscern algebraic structures – this theory is now a familiar feature in physics,chemistry and biology.

10.4. Linear algebra and non-commutative algebra

The concept of the vector dates back to the end of the 16th Century, to SimonStevin. We can also consider that it was Leibniz (in a letter to Huyghens in 1679) whowould develop the earliest rudiments of a calculus using these geometric entities. Allthe same, it was not until 1835 and the Italian mathematician Bellavitis that the firstwork on the calculus of “equipollent” lines (or, as we know them today – vectors) wasseen. It would seem that the origins of linear algebra date back to the 19th Century.In 1839, in his dissertation entitled, “The Ebb and Flow of Tides”, H. G. Grassmann(1809–1877) used these vectorial methods, thus defining the sum and determinant ofvectors in a plane and space. It is thus the origin of a purely combinatorial theorythat is much more general than its application to 3D space would have suggested. Thestructures of a vector space and algebra can, strictly speaking, be traced back to this6.

At around the same period, D. W. Rowan Hamilton was the inventor of the theoryof quaternions, a highly fertile mathematical theory that could be used in differentfields of mathematics and, beyond this, being used most notably in the formalization

5 J. Vuillemin was one of the first French philosophers to take an interest in this theory [VUL62].6 See [GRA 11] and the excellent commentary by [GRA 68, p. 89-101].


of certain parts of quantum physics. Quaternions, which are geometric operatorelements of a vector space and of algebra over the field of real numbers, made itpossible to develop particularly economical methods of writing in linear algebra.Hamilton developed this quaternionic structure with a view to generalizing ordinaryalgebra and the algebra of complex numbers7. A more general manner of presentingquaternion algebra is of presenting it as the Clifford algebra of a regular quadraticspace with 2(a,b)

K dimensions, where a = b = #1 and K = R, and we know that itsgeneralization to four dimensions was independently discovered by P. A. M. Dirac inhis relativistic theory of the electron8.

Let us recall that a set E, equipped with two internal composition laws, denoted,respectively, by + and . is a ring if and only if:

1) (E,+) is an abelian group.

2) (E, .) is a semi-group.

3) . is distributive with respect to +.

The neutral element of the + law is called the null element; the neutral element ofthe . law is the unit element. If it exists, the ring is said to be unitary. If themultiplication is commutative, the ring is commutative.

As we know, there is a structure that is much stronger than that of the ring, inwhich each element will possess a multiplicative inverse, except the null element thatcannot have this. In effect, a ( E, 0 . a = 0 $= 1. Thus, if E is a unitary ring, we canonly find the inverse for the multiplication in E\{0}.

A set (E,+), equipped with two internal composition laws, denoted, respectively,by + and . is a field if and only if:

1) (E,+, .) is a ring.

2) (E\0, .) is a group.

3) If (E\0, .) is an Abelian group, E is a commutative field.

The inverse of an element a according to the multiplication (.) is written as: a&1.

7 The field of quaternions H is today defined as a non-commutative superfield of the field ofcomplex numbers C. H is a vector space over R (the set of real numbers) with four dimensions,admitting a basis {1, i, j, k}, such that i2 = j2 = k2 = "1 (see [BOU 74, p. 3]).8 See [DEH 81, p. 298 , DEH 93, p. 238]. G.-G. Granger [GRA 68, pp. 80–89] has alsocommented on the preface to Hamilton’s Lectures (1853). Also see our text [PAR 12].


The classic example of a commutative field is the set of real numbers R, equippedwith the usual addition and multiplication. An example of a non-commutative field isHamilton’s field of quaternions9.

Yet another example is the set of n!n square matrices, Mn, equipped with matrixaddition and multiplication laws, with matrix multiplication being non-commutative.

This non-commutativity of matrix multiplication has important consequences,especially for quantum mechanics. In quantum mechanics, where vibrating particlesare described by wave functions, we represent physical variables as radiationfrequencies, the coordinates or the quantities of movements of a particle, forexample, through matrices of numbers. If Xj is the matrix of coordinates and Pj isthe matrix of quantities of movement, the product of the matrices, Xj .Pj , defines thekinematic moment, J , of the particle. Moreover, Xj .Pj $= Pj .Xj . We can alsocalculate the difference Xj .Pj # Pj .Xj = ih

2! . Heisenberg’s famous principle isdeduced from this equation and states: !Xj .!Pj 6 h

2! .

A fundamental structure, of especial importance in physics and applications ingeneral, is that of the vector space.

Let K be a commutative field. We say that a set E, equipped with two laws:

– an internal composition law “+”: E2 #+ E is called addition or vector sum;

– an external composition law “.”: K! E #+ E, called scalar multiplication;

is a vector space over the field K if:

1) (E,+) is an abelian group;

2) the law “.” verifies the following properties:

- It is left-distributive with respect to + in E, and right-distributive with respectto the addition of the field K.

- It verifies mixed associativity (with respect to multiplication in K).- The neutral multiplicative element of the field K, denoted by 1, is neutral on

the left for ., that is for all vectors u, v in E and all scalars ), µ in K, we have:

)(u+ v) = )u+ )v ()+ µ)u = ()u+ µu)

9 A quaternion is a quadruplet q = a + bi + cj + dk, where a, b, c, d are real numbers andi, j, k are such that i2 = j2 = k2 = "1. Multiplication is non-commutative in such a fieldas the set Q2 = 1,"1, i,"i, j,"j, k,"k forms a group for multiplication whose table clearlyshows non-commutativity. In particular, it can be clearly seen that:

ij = k $= ji = "k or again : ik = "j $= ki = j, etc.


)µ(u) = )(µu) 1.u = u

We generally designate scalars using Greek letters and vectors using Latin letterswith an arrow above them. A vector space is also often written as v.e.

Examples for vector spaces are:

1) a field is a vector space over itself;

2) if K! is a subfield of K, then K! is a vector space over K!. For example, thefield of complex numbers C is a vector space not only over itself (case 1) but also overR or over Q, which are sets included in C;

3) if E1, E2, ... are vector spaces, then the product E = E1 ! E2 ! ... ! En is avector space;

4) the ring of polynomials for an indeterminate, x and for a coefficient in the fieldK is a vector space over K;

5) the set of applications of a set A in a vector space E is a vector space for thelaws defined by the equalities:

1a ( A, (f + g)(a) = fa+ ga, (If)a = If(a)

Now let E be a vector space over the field K and F be a non-empty part of E. Wesay that F is vector subspace of E if F is stable for the laws of E and if, equippedwith the resulting laws, F is a vector space over K.

Among the remarkable vectorial subspaces of a vector space E, there is the nullvector, E itself, and all the subspaces whose dimensions are smaller in number to E,vector line, vector plane, etc.

For the vector spaces of finite dimension, the incomplete basis theory states thatin a vector space E, any free family of vectors may be completed into a free andgenerative family of E, that is a basis of E, and that from every generative family inE may be extracted a free and generative subfamily. This theorem states, in particular,that every vector space E admits a basis (as the empty family is free and may becompleted by a basis from E). The result showing the existence of this phenomenon,in conjunction with the theorem, by which all the bases in E have the same cardinal,leads to the definition of the dimension of a vector space. The most general form ofthis theorem is the following:

THEOREM.– Let E be a vector space of finite dimension, G a finite generative part ofE, and F a free part of E. There then exists a part H of G\F such that F 3H = /and F 4H is a basis of E.


The concept of the linear applications from one vector space to another makes itpossible to formalize classic geometric operations, as the set of linear applicationsitself constitutes a vector space. We introduce the concepts of linear group, linear(and multilinear) forms, which, through their matrix representations and the dualitybetween vector spaces and quadratic spaces, may be associated with quadratic forms.The diagonalization of matrices, the computation of proper values and proper vectorswill then make it possible to reduce the dimension by preserving information, anessential operation in multidimensional statistics applications (data analysis andcorrespondences).

10.5. Clifford: a philosopher-mathematician

One major consequence of the advances made in geometric algebra was thephilosophical work of William K. Clifford, and his curious, non-Spinozian theory ofthe mind–body parallelism, which was founded on the algebra-geometric conceptthat would later be called “Clifford’s parallelism”10.

Let us recall that Clifford was behind the algebra that today bears his name andgeneralizes the algebra of complex numbers to spaces of any dimension. In his famous1878 article, he defined an algebra of n units e1, e2, ..., en, such that:

e2i = #1 and eiej = #ejei [10.1]

This was a highly productive algebra, with innumerable consequences, and itwould form the base for Dirac’s relativistic theory of the electron, even though Diracnever cited Clifford, who had studied the case of e2i = 1 (the case of the Diraccoefficients) in his 1882 article. We also note that since eiej + ejei = 0, we caneasily deduce from this the equivalence between a quadratic form and the square of alinear form, which is fundamental to Dirac’s theory. Indeed, it is clear that conditions[10.1] make it possible to posit:

(e1a+ e2b)2 = a2 + b2

and, in general:

(e1a1 + e2a2 + ...+ enan)2 =

$(aj)

2

Prior to this article, however, Clifford had worked on the writings of Hamilton andGrassmann and it was in generalizing these that he made his discovery.

10 We have taken the liberty, here, of condensing a much more detailed article [PAR 09].


Let us recall that in 1843, the English mathematician William Rowan Hamilton,had, after several fruitless attempts, managed to find an equivalent for complexnumbers of the type a + ib for 3D space. He called these “quaternions”, which arenumbers of the form:

a+ bi+ cj + dk with a, b, c, d ( R

numbers whose multiplication (.) is defined by the rules:

. i j ki -1 k -jj -k -1 ik j -i -1

An immediate generalization then led him to take, as coefficients a, b, c, d,complex numbers instead of real numbers and he named these complex quaternions“biquaternions”.

In 1871, Clifford then published an article in the Proceedings of the LondonMathematical Society. This was entitled “Preliminary Sketch of Biquaternions” [CLI71] and in it, as van der Waerden [WAE 85, p. 188] summarizes, he introduced twodifferent types of biquaternions, both of which could be written in the form:

q + (r

where q and r are quaternions, while ( commutes with all quaternions.

Clifford assumes, in the first part of his article, that:

(2 = 0

and he uses biquaternions to describe the movements of a rigid body in Euclideanspace. In the second part, he assumes:

(2 = 1

and he uses the second type of biquaternion to describe non-Euclidean movements.Thus, introducing two new units:

. =1

2(1 + () and / =

1

2(1# ()


he shows that:

.2 = .,/ 2 = / and ./ = 0

which, in modern language, signifies that Clifford’s second algebra of biquaternionsis the direct sum of two quaternion algebras.

The space where the above-defined movements take place may be ordinaryEuclidean space. In this case, the torsion is null and corresponds to the situationwhere ( = 0.

On the other hand, in the elliptical geometry constructed by Cayley, which Cliffordknew about, we have the second case that he envisaged and ( = 1. In such a geometry,we encounter the following conditions:

1) The elliptical space is such that there exist a point for each set of coordinatevalues and a set of values for each point, without exception.

2) There exists a system of fixed forms called an “absolute” system (in fact, aquadratic) in which all points and tangent planes are imaginary. If the line joining twopoints a and b meets the absolute at i and j, the quantity ab.ij/

"ai.ajbi.bj = ab,

which is a function of the anharmonic ratio and (thus constitutes an invariable), iscalled the “power of the points a and b with respect to one another”. The distancebetween these two points is an angle $ such that sin $ = ab. This condition can begeneralized to planes.

3) If two points are conjugated with respect to the absolute, their distance is thatof a quadrant. If two lines or planes are conjugated with respect to the absolute, theyform right angles. If any point at a distance of a quadrant from every point situated inthe plane is called a “pole” of that plane, then over the absolute, all the points whosedistance from a given point is that of a quadrant are located in a polar plane withrespect to the absolute. It is generally possible to draw only one perpendicular on agiven plane through an arbitrary point. However, if this point is the pole of the plane,then each line passing through this point is perpendicular to the plane.

4) It is, therefore, always possible to trace two lines on Clifford’s quadratic, suchthat each of them meets two given lines at right angles and that these are respectivelypolar to the first (the two lines mentioned previously). A line can thus be transformedinto another through a rotation around the two polar axes between them. When thesetwo axes are equal, the lines are equidistant, that is they are parallel. But they can beparallel in two ways: we call a line a right parallel when it is the transformation ofanother line through a rotation to the right. A left parallel is a line that results from aninverse rotation. Clifford parallels have no common point and are equidistant, in whichrespect they are close to the classic Euclidean parallelism. However, unlike Euclideanparallelism, these parallels are never situated in the same plane. This last element is a


considerable generalization of the notion of parallelism that would be used by Cliffordin philosophy.

A complete philosopher, Clifford was not content with simply reflecting on theconcept of space or becoming the epistemologist of his discipline. He alsoapproached, in non-mathematical work, different problems in conventionalphilosophy such as problems related to the mind–body relation, or again questionsfrom the field of ethics and social philosophy. These are the aspects of Clifford’sphilosophy that we will explore here.

In his article, “Body and Mind”, reprinted at the beginning of his work Lecturesand Essays, he never likened the mind to a form or substance as did Descartes (andmany of the authors around the 17th Century). He saw it, instead, as a veritable “flowof sensations” that was produced “in parallel” to a certain action of the body [CLI 01,p. 34].

This “parallelism” must, however, be seen in a much more general mannercompared to how Spinoza saw it – purely and simply Euclidean. Based on what hehad developed in “Preliminary Sketch of Biquaternions”, Clifford explained that inan elliptical geometry, like that introduced by Cayley and developed by Cliffordhimself, the parallels that have since been called “Clifford, parallels” lose some oftheir Euclidean properties [BOI 95, p. 448] and only retain a functional character.

Moreover, in an article titled “The Nature of Things in Themselves”, alsopublished in Lecture and Essays, it is exactly this form of parallelism that isdiscussed in the context of the problem of the relation between mind and body.

The word “parallelism”, he explains, signifies here a “parallelism of complexity,an analogy of structure [CLI 01, “The nature of things in themselves”, p. 61].

We must first understand this complexity of the mind. Any sensation that isexperienced is, in itself, complex and is accompanied by an infinite chain ofmemories, which are also complex. These massive organic sensations, whichapparently lack connection with the objective order of things, are what we ordinarilyassociate with the idea of consciousness. This can, further, be easily disturbed byexternal sounds (a barking dogs) or internal parasitic sensations (an incipienttoothache). In this context, consciousness can only be a series of groups of changesthat are related through a set of connections. However, as soon as this link isestablished, a sort of absolute feeling is created. And all the other images are thenrelated to this.

Everything, thus, takes place like in elliptical geometry, where there is an“absolute”, the conics that bears this name, to which all other figures are related.Mathematical entities such as the vector represent instantaneous phenomena (speed,force, power, intensity, etc.) and there are also ordered sets of points (lines) that may


be defined as “parallels” in the sense used by Clifford. Finally, it is also possible todefine transformation groups and entities such as quaternions or, later, spinors wouldbe the very operators that could translate rotations and torsions comparable to thosethat are produced in consciousness during the interweaving of different “threads”.

We thus find elements of Clifford’s geometry in his philosophy. And as wasestablished in the article “The Nature of Things in Themselves”, this is a doubleparallelism within the mind stuff : the conventional parallelism between externalreality and mental image is overlaid by a second parallelism, the parallelism betweencerebral image and phenomena, which, analogous to Clifford’s mathematical work,seems to suggest that these realities are, in fact, what are today called “cliffparallels”[BER 79, p. 67]. As shown by M. Berger [BER 79, vol. 2, p. 196, vol 5, p. 67], wehave a beautiful translation of this in Euclidean space on the sphere or the torus(Villarceau circles), an illustration of which is given in Figure 10.1.

C

c(t)

r(!)

C’

m

m’

Figure 10.1. Clifford parallelism on a sphere and on a torus

In Clifford’s work on ethico-political problems, we again see shades of hisgeometric methods. This work predates the publication of his 1878 article“Applications of Grassmann’s Extensive Algebra”, but he was already familiar withGrassmann’s work and influenced by his scientific work. In the article “On theScientific Basis of Morals”, published in the Contemporary Review of September1875, Clifford posits that a scientific moral must be founded on three principles:

1) hypothetical maxims;

2) principles derived from experiments;

3) the assumption that there is certain uniformity in nature.


According to the mathematician, we must look for the origins of moralsentiments and social conduct in what he called “the tribal self”. This is the part ofthe mind where individuals structure their “gregarious instincts”. This, according tohim, is the origin of our “piety”, a word used in the sense of the Latin pietas, whichalso means “pity”. It is to this sentiment that Jean-Jacques Rousseau traced the originof social conduct. Thereupon, according to Clifford, encouraging pious attitudes anddiscouraging the “impious” was the leitmotif of all societies. The tribal aversion thatwas felt for the delinquent was, essentially, the transposition of the aversion that onemay have with respect to a predator or harmful beast. The process involved informing moral sentiments, therefore, assumes, first of all, a feeling of analogybetween self and others, which assumes that we are able to put ourselves in theirplace; second, the expression of aversion toward something that breaks this analogyand the rectification that follows (applying repressive principles). Consciously orunconsciously, such a process, whose goal is to improve human, is then extended tocomplex social arrangements.

It can be observed here that the mathematician applies moral principles that aresimilar to those that are commonly used in geometry:

1) a principle of symmetry that turns the different “self”; into interchangeablefigures;

2) an operation that makes it possible to transform a deviant (or deformed) figureinto a figure that is perfectly symmetrical to others.

The aim of this operation is clearly not the happiness of each person, but thepossibility of living together and, beyond this, the improvement of each common life.Thus, if we think about it, the operation combines a sort of rotation (returning to theright path) accompanied by an extension (improvement). This is, typically, a sort of“quaternionic” process.

The delicate point in this entire proposition is, of course, the relation betweensociety and the individual, the articulation of the internal and the external. In the textentitled “Right and Wrong: The Scientific Ground of Their Distinction”, Cliffordwould examine this question in greater detail and confirm the idea that societiesprioritize actions that tend to give the community a greater advantage in its strugglefor existence.

Political, religious [CLI 01, “The ethics of religion”, p. 206] or moral idea are thusdefinitively hinged on the imperative need of social improvement just as, elsewhere,with the idea of truth, which is the necessity of not going wrong in a situation wherethe survival of the community essentially depends on the confidence that humans canget along.

In this analysis of individual and collective responsibility, where the changes inthe “tribal self” are carried out with respect to the modifications imposed by society,


we can still undoubtedly find the distant traces of the distinction that Grassmannspeaks about in the beginning of his article, between external or scalar algebra (inwhich the concept of product assumes identical factors) and internal or polar algebra(in which factors play different roles). Clifford’s societal algebra is, in sum,equivalent to the first, in the sense that through its laws it tends to state that allindividuals are equivalent, while the internalization of rules in the “tribal self”transforms each character with a precise goal, social improvement, operating moreby a polar multiplication, analogous to the quaternionic product discussed earlier.

Thus, scientific and philosophical reflections do correspond and the latter isundoubtedly enriched by the illumination, if not formalisms, of the former.

11

Topology and Differential Geometry

Topology is the part of mathematics that studies the qualitative nature of space andthe relative position of points or sets of points that make up this space.

In the 20th Century, the veritable century of topology [DIE 77, p. 1], thisdiscipline has illuminated all fields of mathematics and has become a highlypowerful analytical tool that can be combined with other disciplines, notablydifferential geometry. This thus makes it possible to give very precisecharacterizations of spaces and their continuous deformations, leading toclassifications, especially in the case of surfaces.

Finally, we will see that because of its aptitude for describing any form, topologyhas been applied in numerous fields, from biology to linguistics and passing throughall disciplines that deal with space, from physical geography to spatial analysis.

In order to help readers who may not be mathematicians to understand the laterapplications of topology and the vision of the world that results from this, we firstpresent a very brief introduction to the history of this discipline. We will then introducethe fundamental concepts that are the building blocks of this discipline, even at the riskof making it slightly heavy reading.

11.1. Topology

The history of topology dates back to Leibniz, who was the first to have envisagedthe existence of such a science (related to neither quantity nor measurement, but toquality and the variety of forms and entities). He gave it the name Analysis Situs.

The plan to qualitatively analyze space remained an unfulfilled desire for Leibniz.It was instead the mathematician L. Euler who was behind one of the first results in



topology by resolving the very famous Koenigsberg bridge problem, which is also themost distant known ancestor to graph theory1.

Euler’s work would, however, have no impact for a long time. It was not until the19th Century and Riemann and Listing’s work that topology would be given dueconsideration again. This discipline would then truly develop through Poincaré’sresearch and the qualitative methods he used to study the shape of the trajectories ofdynamic systems.

The essential aspects of the vocabulary of topology would thus be formed onlybetween the end of the 19th Century and the middle of the 20th Century. These centralaspects are discussed in the next section.

11.1.1. Continuity and neighborhood

The qualitative character of space was clearly revealed in mathematics throughstudying the concept of the continuity of a function, which dates back to Cauchy. Toexpress that the graph of a function f(x) does not present any break, we study pointsx that are “sufficiently close” to a given point, x0. This is how the concept of“neighborhood of a point” was introduced. The definition initially still involved ametric concept. But, it was rapidly seen, in 20th-Century thought, that it was notessential to specify the concept of neighborhood for this point in order to definecontinuity.

The initial attempts in this direction were carried out by Fréchet in a 1906 article[FRE 06, pp. 1–74]. This major French mathematician attempted to show what wascommon to properties of sets of points and functions, without bringing in the conceptof distance. A little later, in 1908, Riesz attempted something of the same order. Butneither of these mathematicians was able to construct a practical and productivesystem of axioms. It was not until F. Hausdorf [HAU 06] that general topology as weknow it today was born.

To access this representation of the concept of neighborhood without distance,we must give a precise definition of continuity, which will impose restrictions on thedefinition of neighborhoods.

For example, if f is an application of a subset A of a Euclidean space E in anotherEuclidean space F , then we say that f is continuous at a point p if, for any givenneighborhood U of f(p), there exists a neighborhood V (p) such that f(V 3A) 7 U .

1 See our commentary in [PAR 93a].

Topology and Differential Geometry 255

We immediately have the following consequences for the definition ofneighborhood, which cannot be characterized any which way:

1) The most obvious condition is that every neighborhood of a point p musteffectively contain p.

2) A neighborhood of p, if we do not specify the distance of p from its border,can be more or less extended. As a result, a second condition can be established thatevery subset of a set E, containing a neighborhood of p, may be considered as aneighborhood of p.

3) If f and g are two functions continuous at p, it is valid to assume that their sum,that is, the function f + g, is also continuous. The result is that if U is a neighborhoodof p for f , and V is a neigborhood of p for g, then the intersection U 3 V must be aneighborhood of p for f + g. This presupposes, as a condition for neighborhoods, thatthe intersection of two neighborhoods is indeed another neighborhood.

We can furthermore make the following observation: given that a neighborhood,U , of a point p in Euclidean space (of two dimensions, for example) necessarilycontains a ball U ! whose center is p and which has, let us say, radius r, and because,for any point, the distance of p! from p is strictly smaller than r, we can trace a ballwhose center is p! and whose radius is quite small and entirely contained within U !.The result of this is that U is also a neighborhood of such a point, p!.

However, U !, the set of points whose distance from p is strictly lower than r, is aneighborhood of p. Therefore, we have just shown that a neighborhood U of a pointp in Euclidean space is also a neighborhood of any point p! in a certain neighborhoodU ! of p. It can be seen that this is an intuitive, very natural property of the concept ofproximity, if expressed. Thus, the points close to p are also close to all points that aresufficiently close to p. The natural character of this property of proximity leads to itbeing preserved in an abstract theory2.

We thus access the first definition of a topological space.

11.1.2. Fundamental definitions and theorems

DEFINITION.– [Topological Space] A topological space is an abstract set E in whicheach element p of E is associated with a non-empty family of the subsets of E. Theelements of E will be called points and the subsets associated with a point p in E will

2 On this natural introduction of neighborhoods, see [WAL 57, pp. 9–15].


be called neighborhoods of p. The neighborhoods of each point of E must satisfy thefollowing conditions:

1) If U is a neighborhood of p, then p ( U .

2) Any subset of E containing a neighborhood of p is itself a neighborhood of p.

3) If U and V are neighborhoods of p, then U 3 V is also a neighborhood of p.

4) If U is a neighborhood of p, there exists a neighborhood of p, V , such that U isthe neighborhood of every point in V .

We sometimes say that E is equipped with a topology, or that we define a topologyover E, when every element of a set E is associated with neighborhoods such that Ebecomes a topological space.

Starting from the definition of neighborhoods in a topological space, it is possibleto give a mathematical meaning to an entire series of terms that have a very strongintuitive significance. We thus define the concepts of interior3, openness4, limit5 andlimit point6, closedness7 and boundary8.

3 Let A be a set of points in a topological space E. We say that a point p in A is an interiorpoint of A if there is a neighborhood U of p such that U % A. We call this “interior to A”, anddenote by I(A) the set of all points interior to A.4 A set A of a topological space is called an open set if, for each point p & A, there exists aneighborhood U of p such that U % A. From this definition and that of neighborhoods, wecan deduce that the union of any family of open sets is an open set, that the intersection of afinite family of open sets is also an open set, that the space E is completely an open set and thatempty set (') is also an open set.5 We say that a series of points p1, p2, ... located in a topological space E has the limit p, orconverges at p, if, regardless of the neighborhood chosen U(p), there exists a whole numberN such that pn & U for all n ( N . A topological space in which the limit of any series isunique is a Hausdorff. In such a space, for any pair of points p, q with p $= q, there exists aneighborhood U(p) and a neighborhood V (q) such that U ) V = '. We say that such a spaceis “separated”.6 Let A be a set of points in a topological space E. We say that a point p in E is a limit point ofA if every neighborhood of p contains a point in A different from p.7 Let A be a set in a topological space E. The subset of E constituted by all of the points of Aand all limit points of A is called a closed set in A, denoted by F (A). E is closed if and only ifthe complement of A in E is open.8 Let E be a topological space and A be a set in E. Then, the boundary of A is the set of allof the points in E that belong neither to the interior of A nor to the interior of the complementof A, denoted by CA. From this, we easily deduce that F(A), the boundary of A, is equal to:F (A) ) F (CA).


11.1.3. Properties of topological spaces

The above elements make it possible to compare different topological spaces undercertain conditions and, if necessary, make it possible to identify several forms that donot have the same appearance or the same dimensions with the same object.

We sometimes say that a topologist is a mathematician for whom there is nodifference between an elephant and a teacup. Indeed, we will see that apparentlydissimilar shapes can be topologically equivalent, up to deformation. To define thevalid deformations that make it possible, in some measure, to classify topologicalspaces into families of equivalent objects upon transformation, we define an initialset of very evident properties of topological spaces: the concepts of homeomorphism,compactness, path, arc-connectedness and connectedness.

The concept of homeomorphism between two topological spaces E and F isbased on the idea of continuous, reciprocal bijective applications of E on F and of Fon E, which is the same as correspondence by deformation. This is the centralconcept in topology but, at the same time, it is a very strong property and is onlyrarely verified. Thus, the majority of classifications of topological spaces are basedon weaker properties.

We next have the idea of a cover (a family F of non-empty subsets of a set Ewhose union contains the given set E) and open cover (all of the sets of F are opensets), and from this we derive the idea of compactness: a topological space E is acompact space if: (1) this is a Hausdorff space; (2) any open cover of E contains afinite subcover. In this case, we also say that the space is closed and bounded.

The concept of path is again defined non-intuitively. It is not formed by a set ofpoints by the applications that make up the set9, and the different concepts ofconnectedness (which translated the idea of a “single tenant” entity):arc-connectedness10 or, a weaker property, just connectivity11.

9 Let E be a topological space and I be the unit interval 0 * t < 1, considered to be a subset ofthe space of real numbers for the usual topology. By definition, a path in E joining two pointsp and q in E, is a continuous application f of I in E, such that f(0) = p and f(1) = q. We saythat the path is contained in a subset A of E if f(I) % A.10 We say that a topological space E is arc-connected if, for any pair of points in E, p andq, there is a path in E joining p and q. If A is a subset of a topological space E, then A isarc-connected if any pair of points in A can be joined through a path in A.11 We say that a topological space is connected if it is not possible to represent it as the unionof two open, disjunct sets, both of which are non-empty. A subset A of a topological space Eis said to be connected if, when considered as a subset of E for the induced topology, A is aconnected space.


Compactness, arc-connectedness and connectedness are properties of topologicalspaces that are inherited through homeomorphism. These properties are, in fact,elementary. We will now examine properties that are far less immediate, whichessentially result from the possibility of associating a topological space with acollection of groups – groups assigned to homeomorphic spaces being isomorphic.We then say that these groups, which make it possible to identify mathematicalobjects that differ in appearance, make up topological invariants.

For example, to differentiate between a disk and a ring, we can examine thepossibility or impossibility of contracting the closed paths traced in these two spacesto a single point. This contraction is seen to be possible in the disk, but not in thering. By giving this possibility a precise meaning, we have the concept ofhomotopy12.

The relation of homotopy is a relation of equivalence over the set of all paths in theconsidered space based on the same point. We can thus partition the set of these pathsinto homotopically equivalent classes. It is therefore possible to define the product oftwo homotopic classes (related to the product of two paths), as the introduction of thisoperation produced on the classes of homotopy makes it possible to turn this into agroup. We thus show that if two spaces connected by arcs are homeomorphic, thentheir homotopic groups are homomorphic.

Other topologically invariant groups may be attached to topological spaces. Toattain these, we can begin with the empirical observation that over topological spaces,certain curves make it possible to divide the space into two, separate regions, whileothers do not.

For example, it is clear that the possibility of tracing a closed curve over a surfacewithout dividing the surface into two separate regions is a topological property thatcan be expressed as follows: if S is a surface and C is a curve traced over this, and ifS! is homeomorphic with S and if C ! is the curve of S! that corresponds to C, then Cdoes not divide S if and only if C ! does not divide S!.

This topologically invariant property thus provides a criterion that can be usedto differentiate between topologically distinct surfaces. However, this is clearly not avery fine criterion as there are many different topological surfaces on which we cantrace one or more closed curves of the above type.

12 Let I be the unit interval made up of real numbers, s, such that 0 * s * 1. We say that aclosed path f in a topological space E, beginning and ending at x, may be contracted at x or ishomotopic with respect to a basepoint x if f is homotopic with respect to a basepoint x to theconstant application e : I "+ E defined by e(s) = x for all s belonging to I .


This would suggest that we attach a numerical measure to the properties ofseparation of closed curves on a surface. In effect, it is clear that the maximumnumber of closed curves along which we can cut the surface without dividing it intotwo, or more than two, separate regions will be a topological invariant. This invariantmakes more precise the property of the existence (or not) of curves that areboundaries of portions of the surface. All of this is closely related to more elaborateinvariants and suggests that studying the way in which the closed curves on a surfacelimit regions, provides a test that may help in distinguishing between topologicallydifferent surfaces. In fact, we can show that the closed surfaces may be completelyclassified using this test.

If we generalize what we have just discussed about surfaces to other spaces, wemust examine not only whether closed surfaces are boundaries or not, but also whetheror not the pieces of the closed surfaces (of 2-, 3-, ..., r-dimensions) are boundaries ofanything.

We are thus led to precisely define what we mean by an r-dimensional surfaceplunged into a space and also what we mean by boundary. We thus happen upon theconcepts of a p-dimensional simplex in a Euclidean space and the homology group.

DEFINITION.– Let x0, x1, ..., xp, p + 1 be linearly independent points in ann-dimensional Euclidean space. Then, by definition, the Euclidean p-simplex[x0x1...xp] is the set of all points whose coordinates (z1, z2, ..., zn) verify:

zı =n$

&=1

)ixij (i = 1, 2, ..., n) (j = 1, 2, ..., n)

with:

)i 6 0 and$

)i = 1

The geometric significance of this definition is as follows: the numbers )i are, infact, the “barycentric coordinates” of the point corresponding to the simplex, that is,the coordinates that accept the origin as the location of “barycenter” of all the points.

DEFINITION.– We will now say that the standard Euclidean p-simplex, !, is thesimplex of the (p + 1)-space Ep+1 whose vertices are the points (0, 0, ... ,1, ... , 0)with unity at the ith place, for i = 1, 2, ... , p+ 1.

This definition makes it possible to rigorously determine the idea of plunging asimplex in a given space. We can then precisely define a singular p-simplex:

DEFINITION.– A singular p-simplex, or singular simplex of dimension p in a spaceE, is, by definition, a continuous application * of !p in E.


We can thus now distinguish the singular simplex as an application, denoted by(x0, x1, ...xp), of the simplex considered as a set of points, denoted by [x0, x1, ...xp].

This concept of a singular p-simplex is the building block from which we canreconstitute p-dimensional surfaces. The operation that consists of sticking togethersimplexes to form portions of the p-dimensional surface of the space E consists, infact, of “adding” these simplexes. This cannot be done without the introduction of aspecific algebraic structure. The most natural way of realizing this is constructing anadditive abelian group whose generators (infinite in number) are singular p-simplexesof E, called “p-chains”.

As we now algebraically represent the fact of sticking together the portions ofspace by forming linear combinations of singular simplexes, we must define theoperation of forming the edge of a singular simplex. We must define an operationover (x0, x1...xp), which, first of all, translates the geometric fact that the “edge of[x0, x1...xp]” set is the union of its faces and which then becomes a simple algebraicrule that is easy to work with. Each simplex must appear only once in the edge andwe must not, therefore, count the same face twice. Finally, as a closed surface has noboundary, it is natural that it is required that this operation, d(x0x...xp), be equal tozero. Some examples of this in smaller dimensions show that all of this is realizable,with the right formula being:

d(x0, x1...xp) =p$

i=0

(#1)i(x0, x1...,xi, ..., xp)

We then verify that:

d2(x0, x1...xp) =p$

i=0

(#1)i(x0, x1...,xi, ..., xp) = 0 [11.1]

an operation that simply indicates that the boundary of the boundary is empty13.

We can then introduce the following definitions:

DEFINITION.– We call a p-chain + such that d+=0, a p-cycle of a space E.

DEFINITION.– If + and 0 are two p-cycles in E such that + # 0 = d1 for a certain(p + 1)-chain, we say that + and 0 are homologous cycles on E and + is said to

13 We note that this result is related to Michel Gondran’s results for classification of whichthe associated ultrametric matrices, denoted by Dij , are such that D2 = 0. The result is thata classification, and, a fortiori, the meta-classification that is the basis for all classifications,is nothing but an edge when we retain all of the information that it contains (see [PAR 13,PAR 14]).


be homologous to 0. We write this relation as + ) 0. In particular, if + is null and+ = d1, we say that + is homologous to zero and we write + ) 0.

DEFINITION.– Classes of homology, of dimension p over E, are those classes ofequivalence of p-cycles in E for the homology relation.

The chief point of interest of the idea of classes of homology results from thealgebraic considerations. In effect, the group of p-cycles on E, Ap(E), is an additiveabelian group and the relation of homology divides the subgroup Bp(E), ofp-boundaries on E, into modulo classes: two elements, + and 0, in Zp(E) are in thesame modulo class Bp(E) if and only if + # 0 is in Bp(E), which is exactly thesame as the homology relation. Consequently, we can consider the classes ofhomology of the p-cycles in E as the elements of the quotient groupHp(E) = Zp(E)/Bp(E).

We thus arrive at the following definition, with which we will end:

DEFINITION.– The group Hp(E) is called the p-dimensional homology group, ormore simply, the pth homology group of E.

We show that there exists a class of spaces for which it is possible to developa systematic algebraic method to calculate homology groups: simplicial complexes,that is, the subspaces of the Euclidean space formed by a set of points that is the unionof a finite number of Euclidean simplexes (not necessarily all of the same dimension)with the property that the intersection of two of these simplexes is either empty ora “face” for each (the word “face” being used here in its general sense, and not inthe sense of “face of maximum dimension”). We will set aside, here, the study ofsimplicial complexes and the calculation of their types of homology.

11.1.4. Philosophy of classifications versus topology of the being

Very general and highly important concepts in topology (neighborhood, interior,openness, closedness, compactness, connectedness etc.), which may be applied tocharacterize all sorts of sets and spaces, also find diverse applications in differentdisciplines, notably physics and cosmology. But, they can also lead to notablephilosophical extensions if we draw conclusions from their concepts and use them ina precise and rigorous manner.

Knowing that a hierarchical classification corresponds to an ultrametric matrix ofdistance Dij , such that D2 = 0 for an operation *, interpreted as the “min”, and theoperation 8, interpreted as “max”, we cannot avoid approaching such a classificationof a p-simplex associated with the concept of “boundary”. If we then accept that theuniverse can be likened to the elaboration of an immense hierarchical tree-structure,


it results that this structure, contrary to certain hypotheses postulated by S. Hawking,must necessarily include a boundary.

We can conceive of philosophy, this “topology of being”, as Heidegger says (usingthis word in a purely poetic sense, however), as an informal method of recollementof different simplicial complexes that make up the space of knowledge; metaphysics,the quest for origin, can be seen as the absolute search for a sort of “boundary of theworld”.

Most philosophers have not gone this far, in general. The uses of topology inphilosophy, when they exist (e.g. Bergson never made use of them even though hewould have greatly benefited from them!), have remained extremely metaphorical, aswith Bachelard. In his Poetics of Space, Bachelard undertook an analysis of the sitesof intimate life, which does indeed refer to the Analysis situs, but which makes use ofnone of its technical concepts. The rest need not be discussed here14.

11.2. Models of differential geometry

In this section, we do not intend to comprehensively present all of the features ofdifferential geometry, and will notably not be approaching the set of available modelsthat make it possible to explore different types of spaces at work not only in geometrybut also in physics and, more generally, in any discipline where space is involved. Welimit ourselves to a short reflection on space as a support for thought. We then highlightthe importance of differential geometry and present how René Thom used this in hisfamous “théorie des catastrophes” (theory of catastrophes), a truly neo-Aristotelianphilosophy of nature. We hope that the discussion will be comprehensible to non-mathematicians.

11.2.1. Space as a support to thought

We can, of course, only think in space. To think is to reestablish or modify existingrelations between concepts that are, in reality, classes. For example, if we say thata whale is a mammal, we are implicitly establishing a relation of inclusion betweenthe class of whales and the class of mammals. These relations may or may not possess

14 Deleuze and Guattari’s arguments, in particular, in Qu’est-ce que la philosophie? (What isPhilosophy) to try and define a philosophical space, are absolutely worthless for anyone whoknows a little bit about the subject. Equally questionable (as they were difficult to appreciate inthe total absence of any associated formalism) are numerous uses of the word that are scatteredthrough Michel Serres’ flamboyant work and, even when suggestive, can barely be proved to bepertinent.


different properties: for example reflexivity, symmetry, transitivity. A whale is a whaleand a mammal is a mammal. But if a whale is a mammal, it is not necessary thata mammal is a whale. On the other hand, if a whale is a mammal and a mammalis a vertebrate, then a whale is necessarily a vertebrate. The “is a” relation, whichis reflexive, antisymmetric and transitive, is a relation of a (large) order, which istranslated by a hierarchical boxing of classes within one another.

As can be seen, relations of this sort make it possible to organize thought bydefining orders, but also, for instance, through equivalences or resemblances betweenclasses. It is also possible to consider some of these relations not as relations definedover a unique set but as applications of a set in another. Some of these applicationsmay then be functions that translate the unique particularities of the correspondencebetween these sets. For example, the fact of being injective, surjective or bijective.

Relations, like functions, may be traced on a graph that represents thecorrespondence established between the terms or sets that have been related.

The organization of thought into concepts and conceptual relations is, thus,always associated with spatial representations. We will, moreover, see that whenthese relations define a characteristic structure on sets (e.g. an order structure) orwhen the set of applications of a set in another possesses a well-defined algebraicstructure, it is possible to relate these order structures or algebraic structures totopological structure. These are dual translations of the algebraic or order structuresand also provide proper spatial interpretations of these.

It follows from this that space underlies all possibility of thought. In this sense,geometry is deeply involved in thought or, at any rate, logical thought. We cannotreally object to the proposition that geometry, at least initially, only concerns spacesthat are continuums. In reality, it is perfectly possible to define geometries usingdiscrete supports in such a way that the existence of absolute discontinuities in the setof concepts of classes cannot invalidate our statement.

Spinoza’s idea, according to which it would be possible to treat the affections of thesoul as though they consisted of lines, surfaces and solids, cannot thus be abandoned.It remains that the concept of space, like the concept of geometry, must be refined inorder to be able to provide some sort of support in modeling thought.

11.2.2. The general concept of manifold

Hypothetically, any space can take any shape. Conventional geometry hasprivileged certain shapes in one, two or three dimensions. The generic concept ofmanifold, resulting from Riemann’s work, makes it possible to gather together withinone class the characteristic topological properties of conventional geometric objects


(lines, surfaces, volumes, etc.) and to designate these by the same concept,independent of their dimension. The manifolds that present edges are polygons (twodimensions), polyhedra (three dimensions) or, more generally, p-dimensionalsimplexes. Manifolds that are sufficiently smooth are differentiable manifolds. Anintermediate category would be that of the polytopes.

Another, more topological criterion for classification consists of distinguishingbetween “separable” spaces (including Hausdorff separable spaces) and“non-separable” spaces.

In manifolds that are sufficiently smooth, and thus differentiable, and in the class ofthose that are also Hausdorff manifolds, we can more particularly consider manifoldswhere it is possible to give meaning to the length of a vector of the tangent space:these manifolds are called “Riemannian” manifolds. They can then be equipped with aRiemannian vector or one of its generalizations (pseudo- or semi-Riemannian metric,sub-Riemannian metric, Finsler metrics).

But there are also complex manifolds, Kähler manifolds, which are non-geometricand are also called algebraic varieties.

11.2.3. The formal concept of differential manifold

Analysis is not restricted to producing local results, valid in the neighborhood of apoint or a subset. Thus, we do not study only the behavior of solutions of differentialequations with partial derivatives or integrals in open numerical spaces of Rn. Thisstudy is now carried out through a global analysis and, therefore, over differentialmanifolds that are recollements of local open sets.

A few facts must be established beforehand in order to arrive at the current formalconcept of a differential manifold. We begin by recalling the definition of topologicalvector spaces and Banach spaces.

DEFINITION.– [Banach space] A topological vector space E (over the field of realnumbers R) is a vector space equipped with a topology such that the operations ofvector addition and multiplication by a scalar are continuous.

The most important type of topological vector space is that which we call a“Banachizable” space. This is a complete vector space of which the topology can bedefined by a norm. When the norm is in the structure, such a space is called a Banachspace. Intuitively, the concept of manifold is thus obtained very simply by startingfrom the principle that we can stick together the open Banach subsets withCp-morphisms (morphisms that are p-times differentiable). The formal manner inwhich these recollements operate assumes several steps.


DEFINITION.– [Concept of “atlas”] Let X be a space. We say that an atlas of classCp (p 6 0) over X is a collection of couples (Ui,!i) (i describing a certain set ofindices), which satisfies the following conditions:

1) Any Ui is a subset of X and the Ui covers X .

2) Any !i is a bijection of Ui over an open subset !iUi of a Banach space Ei, andfor every i, j,! i(Ui 3 Uj) is open in Ei.

3) For any couple i, j, the application:

!j!&1i = !i(Ui 3 Uj) + !j(Ui 3 Uj)

is a Cp isomorphism.

DEFINITION.– [Concept of “map”] Any couple (Ui,!i) is called a map of the atlas.If a point x of X belongs to Ui, then (Ui,!i) is said to be a map in x.

DEFINITION.– [compatibility of “atlases”] Let us take an open subset, U , of X anda topological isomorphism ! : U + U ! over a U !, which is an open subset in theBanach space E. We say that (U,! ) is compatible with the atlas (Ui,!i) if eachapplication !i!&1 (defined over a suitable intersection as in the definition of theconcept of “atlas”, condition (3) (see above)) is a Cp isomorphism.

Two atlases are said to be compatible if each map of one is compatible with theother atlas. We can immediately verify that the compatibility relation is a relation ofequivalence and is reflexive, symmetric and transitive over the atlas.

DEFINITION.– [concept of manifold] A class of equivalence for the atlas Cp over Xdefines a structure of Cp-manifold over X . If all of the vector spaces Ei of an atlasare topologically isomorphic vector spaces, we can always find an equivalent atlasfor which they will all be equal to the space E, for example. We then say that X is anE-variety or has E as model. And if E has n-dimensions, we can also say that X isan n-manifold.

11.2.4. The general theory of differential manifold

The general theory of differential manifold poses different problems:

1) The first problem is that of comparing manifolds. That is, mathematicallyspeaking, the applicability of one over the other. This problem is approached byconsidering differentiable applications and the study of the singularity of theseapplications.

2) The second problem is the study of the fields of the tangent vectors. Thedifferential manifolds are often associated, especially in physics, with vector fields.


For a given vector field of a manifold, the consideration of points where it iscancelled has been seen to play an important role in the study of the integral curves ofthis field, of which they are the singular points. It was Poincaré who first discovered arelation between the critical points in a vector field on a surface and the topologicalinvariants of this surface. And it was Hopf who gave this relation its most generalform, which can be stated as follows: let M be a compact variety and X be a vectorfield whose critical points are finite in number. Each point is intrinsically attached toa rational whole number which we call the index of the point. The sum of indices,also called the index of the field X , is the Euler–Poincaré characteristic of M .

In a case where there are k vector fields X1, ..., Xk over the manifold M , thesingular points of the system are the points x ( M where the k vectorsX1(x), ..., Xk(x) are linearly dependent. Moreover, it is possible to generalize theconcept of index to such systems.

A widely studied problem is that of determining the largest entity k for which thek vector fields, X1, ..., Xk, have no singular point. When k = n = dim(M), we saythat the variety M is parallelizable. The problem is entirely resolved for the sphereSn, and we can show that the only parallelizable spheres are S1, S3 and S7.

11.2.5. G-structures and connections

In the 19th Century, as the differential study of surfaces was developing,Ribaucour and Darboux perfected a specific method for this purpose, that of the“mobile trihedron”. In the early 20th Century, Elie Cartan greatly widened the scopeof this method by skillfully applying it to various questions in differential geometryand the general theory of equations with partial derivatives. Finally, C. Erhesmannclarified and systematized Cartan’s ideas by inserting them into the theory of fiberbundles.

DEFINITION.– [Fiber Bundles] Let us recall that a bundle is any tripletX = (E,", B), where E and B are two topological spaces and " : E + B is acontinuous application of the surjective rule. The space E is the total space of thefiber B and its base space. A bundle with base B is said to be a bundle “on” or“above” B." : E + B is the projection of X and the reciprocal image "&1(b) ofany point b ( B constitutes the fiber of X above b.

DEFINITION.– [Vector bundle] A triplet X = (E,", B) is a vector bundle ifX is a bundle and if: (1) for any b ( B, the bundle Eb = "&1(b) has aBanach space structure; (2) given {Ui}, an open cover of B, the applications'i : "&1(Ui) + Ui ! Ei (where Ei is a Banach space) satisfy the followingconditions:


1) 'i is a commuting isomorphism with the projection on Ui, such that thefollowing diagram is commutative:

"&1(Ui)%i

!!

""!!!

!!!!

!!Ui ! Ei

##"""""""""

Ui

2) For any b ( Ui, the application carried out on the bundle (denoted by 'i(b)) issuch that:

'i(b) : "&1(b) + Ei

which is an isomorphism of topological vector spaces.

3) If Ui and Uj are two elements of cover, the application of Ui 3Uj in L(Ei, Ej)given by:

b + ('j'&1i )b

is a morphism.

When these conditions are united, we say that the set {(Ui, 'i)} is a trivializingcover of " (or of E, through a misuse of language). Two such covers are said to beequivalent if the “union” cover satisfies the three conditions stated earlier. We then saythat an equivalence class of trivializing covers determines a structure of vector bundleover " (or over E, through a misuse of language).

DEFINITION.– [tangent bundle] Let M be an n-dimensional differential manifold andTM be the manifold of vectors tangents on M . Now, let " : TM + M be thecanonical projection associating each vector u ( TM with its point of contact of thetangent p ( M , that is, a point such that u ( TpM . By definition, the projection "has as its fiber "&1(p), and we prove that the triplet T (M) = (TM,",M) is a vectorbundle of rank n, called a tangent bundle on the manifold M . Such a bundle has, asfiber, "&1(p), the tangent spaces to the variety M , TpM .

Let us now introduce the concepts of principal bundle, frame bundle andG-structures.

DEFINITION.– [group action, orbits of a group, set of orbits] Let us recall that a groupG is said to “left-operating” on a set E (or said to be the transformation group), ifwe define an application of G ! E + E, such that any couple (a, p) in GE, where


a ( G, p ( E, corresponds to the product ap in E, such that we have, for any a, b ( Gand p ( E:

(ep) = p, a(bp) = (ab)p

with e as the unity of the group G.

A right-hand action of G in E is defined in a similar way.

We see that this action of G defines a relation of equivalence (9) over E, suchthat p 9 q if and only if we find a a ( G, for which q = ap. The correspondingclasses of equivalence constitute the orbits of this action. Any two orbits are eitherconfounded or distinct. An orbit containing p ( E is denoted by Gp and formed byall the elements ap, a ( G. The set of all orbits is designated by the quotient E/G.

DEFINITION.– [principal bundle] Let X = (E,", B) now be a fiber bundle, whereB = E/G is the set of all orbits of the group G that operate on the left of the set E (or,as we have seen before, make up its “transformation group”). X is called a principalG-bundle of the structural group G and the space E (equipped with a free right-handaction on G) is a principal G-space.

DEFINITION.– [frame bundle] Let us now associate a tangent bundle T (M) of avariety M with a principal fiber bundle R(M), called the “frame bundle” of M . Atevery point x in M , the fiber is then the set of bases of the tangent space Tx(M) andthe linear group Gl(n,R) is the group of this principal fiber.

T (M) may then be considered a vector bundle of the structural group GL(n,R).If G is a closed subgroup of Gl(n,R), a G-structure over M is a subspace SG(M ) ofR(M), which is a principal bundle with group G and base M (for the action of G onthe fibers, which is a restriction of the action of Gl(n,R)). T (M) then appears to bea vector bundle of the structural group G and it is usual, since Cartan and his mobileframe method, to work in SG(M ) rather than in T (M) to benefit from the richeststructure. The G groups may have several aspects:

1) G = O(n), orthogonal group. In this case, the G-structures are calledRiemannian structures.

2) n = 2m is even, and G = Sp(2m,R) is a symplectic group. We thus definesymplectic structures (or almost-Hamiltonian structures).

3) n = 2m is even, G = Gl(m,C) is the complex linear group. G-structures arethen said to be almost complex.

11.3. Some philosophical consequences

Among the mathematician-philosophers who were greatly influenced by the workthat we just discussed, we cite three here: A. N. Whitehead, who was first a logician


and Russell’s associate, before entirely becoming a philosopher; A. Lautman, who wasinitially a philosopher but whose mathematical culture was, in his time, unrivaled;and finally, a true mathematician R. Thom, whose work led him to develop a trulytopological vision of the world, which could never have been conceived of withoutthe concepts introduced previously.

11.3.1. Whitehead’s philosophy and relativity

The fact that physics gave greater importance to certain manifolds (Riemannianmanifolds in four dimensions, equipped with a Riemannian metric) played animportant role in the development of Albert Einstein’s theory of relativity. Thestructure of space–time is that of a pseudo-Riemannian manifold. In other words, thatof a G-structure, where G is the Lorentz group that leaves invariant the quadraticform x2

0 + x21 + x2

2 # x23 (where x3 = ct) in R4. The theory of the geodesics of such

a structure, and that of the singularities of differential applications, would then playan important role in relativistic cosmology.

A philosophical extrapolation of these results emerges from A.N. Whitehead’sphilosophy, which was contemporaneous with the birth of the theory of generalrelativity. As Jean Ladrière observed, the theory described by Whitehead in Processand Reality is entirely inspired by the conceptual framework of the theory ofrelativity, notably the fundamental concept of “field”, introduced into physics byMaxwell, and which became the primordial idea of relativity.

In the theory of relativity, the universe is not made up of things but of eventsthat are at the meeting point of two lines in the universe; every point in space–timecorresponds to a double-cone (the Minkowski cone) whose vertex is located at thispoint. The lower cone represents the past and the upper cone represents the future.All of the points of the past may be the origin of an action that can influence, at thepresent moment, the given point. At the same time, the points located outside of thedouble cone do not have a direct causal relation to it. “There is no distant instantaneousaction; the actions are propagated with finite speed, but it is perfectly possible thatlines in the universe that are very far away from each other in the present may meet inthe future. Space–time, envisaged in its totality, thus forms a sort of tight fabric of thecriss-crossing lines of the universe; it is like an immense field of possible interactions”[LAD 71].

Based on this theory, Whitehead then wanted to “represent the world as a singlegeneralized field and to explain all appearances by the properties of this field.Moreover, a physical field is a sort of universal potentiality: it is, simultaneously, thesite of emergence of phenomena, the law governing their production as well as thesystem of their interrelations. If we follow the implications of this idea to the end, wecan no longer think of the world in terms of substance, but we must think of it in


terms of events and the propagation of interactions” [LAD 71, p. 276]. Moreover,what Whitehead calls an “actual entity” is precisely the “integration of all itsprehensions, that is, the total system of all its interactional relations with all others. Itis a node of lines, not an autonomous subject that carries characteristic properties ofitself” [LAD 71, p. 277]. The concepts of relativity are, of course, constructed withinthe statements of theoretical physics. In this sense, they are the object of purelyoperational understanding, which is alien to philosophy. However, they have inspired,within philosophy, the construction of another conceptual field defined in relation toa project of totalization.

We thus find, in Whitehead’s metaphysics, a transposition of this schema because,as he explains himself, “it is in the nature of a being to be a potential element of anyfuture” [WHI 69, p. 72]. Constructed on the basis of 45 categories, whoseinteractions he describes in great detail, Whitehead’s system establishes a vision of auniverse “that is a process, not a system of things, a fabric of encounters, not anetwork of substances, an intertwining of events, not an order formed of natures”[LAD 71, p. 275]. Three categories are particularly significant: actual entities,prehensions and nexa. Actual entities are “the ultimate things of which world ismade”; prehensions are characters of those through which they open up to all othersand take place in a sort of universal interconnection; finally, nexa are the phenomenaof co-occurrence in which actual entities are involved. These indications are enrichedby the introduction of other categories and must, in particular, be included withrelation to the category of the ultimate, which is made up of three concepts – unity,multiplicity and creativity. This last concept is a sort of unifying power introducingnovelty into the world and producing new forms of co-occurrence that Whiteheadcalls “concrescence”: each day, the world becomes more and more “concrete”, in itsetymological sense (Latin cum-crescere, to grow together) and rises to greater andgreater concretion. “Each grain of existence is increasingly loaded with all othergrains of existence, the fabric tightens more and more, the universe integrates itselfmore and more, as Whitehead expresses, when talking of the consequent nature ofGod, who is none other than the ‘objectivation of the world in God’ [WHI 69,p. 406], the prehension of all actual entities in himself” [LAD 71, p. 275]. Based onthe theory of relativity, Whitehead thus constructed a reticular metaphysics, whichalso opened the door to a new theology.

11.3.2. Lautman’s singular work

Despite the interest in his work, which was developed before the Second WorldWar, Lautman should not really feature in our book. He is more a philosopher ofmathematics than a philosophical mathematician. If, despite this, he is included here,it is for one reason alone: the ideal, almost-Platonic dialectic that Lautman identifieswithin mathematics may, as he suggested, be mathematized itself.


Rejecting the positions adopted by Russell, of the early Wittgenstein and Carnap,all of whom wanted to liken mathematics to logic or to a formal language, Lautmandeveloped a project which he expressed as follows:

“It seemed to us however, that it was possible to envisage other logicalconcepts that could also eventually be related to one another within amathematical theory and which are such that, contrary to earlier cases,mathematical solutions of the problems that they pose could contain aninfinity of degrees. Partial results, approximations that stopped half-way,and attempts that still seem to be blind fumbling, came together, unifiedby a single theme, and revealed in their movement a liaison that isestablished between certain abstract ideas that we propose callingdialectics. And so, mathematics, and especially modern mathematics,algebra, group theory and topology, a jumble of constructions thatmathematicians would be interested in, seem to also contain a hiddenstory, one that is told by a philosopher” [LAU 77, p. 28].

Let us first recall the essentials of Lautman’s work. That which we could call “themathematical realism of Albert Lautman” is founded on two fundamental concepts:the concept of structure and the concept of genesis. Mathematics is, first andforemost, a set of structures that are defined by properties of invariance. For example,as Lautman explains, “geometry in the sense used by Klein is the study of theproperties of figures that are conserved when the space, taken as a whole, is subjectto a certain transformation, forming what is called a transformation group” [LAU 77,p. 34]. Euclidean geometry is thus directly related to the invariance of thedisplacement group; affine geometry (i.e. the study of transformations that establishcorrespondence between a point in a plane or in a space to another point in the planeor space) is directly related to the invariance of another group, the group of bilineartransformations. As for projective geometry, it is related to the invariance of the“group of homographic transformations”. An example for a homeographictransformation that forms a group is the degree of an algebraic curve. Anotherexample is the ratio called the “anharmonic” ratio or double ratio between four pointson a line. Once these structures are discovered, of course, we must still reflect ontheir relations. Lautman clearly saw the problem that would be resolved much laterby the theory of categories.

When these structures are described in detail, different types of organization arerevealed. “The solidarity of the whole and its parts, the reduction of the properties ofrelation to intrinsic properties, the passage from imperfection to the absolute – all thesewere so many attempts at structural organization that conferred upon mathematicalentities a movement towards completion by which we could say that they exist”. Thiswas the conclusion of the first part of Lautman’s proposition.


Other types of schema do exist in mathematics: “Certain mathematical geneses,however, do not lend themselves to description through schemas of this type. Theyobey more complicated schemas where the passage from one genre to another requiresthe consideration of mixed intermediaries between the domain and the desired entity.The mediating role of these mixes is a result of the fact that their structure still imitatesthat of the domain onto which they are overlaid, while their elements already belongto the genre of entities that will arise from this field” [LAU 77, p. 106].

Thus, following the movement of mathematical ideas closely with the theoriesthemselves, Lautman highlights the existence of a “dialectic” of mathematics, whichis undoubtedly anterior to mathematics itself, ontologically speaking, as it results froma “concern” in the Heideggerian sense, or from a rational inquiry of nature in order toobtain responses to questions that may arise in relation to it.

But “while the mathematical relations describe the connections that in fact existbetween distinct mathematical entities, the Ideas of dialectical relations are notassertive of any connection whatsoever that in fact exists between notions. Insofar asposed questions, they only constitute a problematic relative to the possible situationsof entities” [LAU 77, p. 211].

The particularity of such a dialectic, which thus defines an order of theproblematic, is that it is both immanent and transcendent. As Lautman writes,“Insofar as posed problems, relating to connections that are likely to support certaindialectical notions, the Ideas of this Dialectic are certainly transcendent (in the usualsense) with respect to mathematics. On the other hand, as any effort to provide aresponse to the problem of these connections is, by the very nature of things,constitution of effective mathematical theories, it is justified to interpret the overallstructure of these theories in terms of immanence for the logical schema of thesolution sought after. An intimate link thus exists between the transcendence of Ideasand the immanence of the logical structure of the solution to a dialectical problemwithin mathematics. This link is the notion of genesis which we give it, at least as wehave tried to grasp it, by describing the genesis of mathematics from the Dialectic”[LAU 77, p. 212].

Although it does indeed seem as if for Lautman, the dialectic of ideas is thusmetaphysically primordial [BEN 10]; in the case of Plato, this dialectic is found to belargely immanent to the Pythagorean mathematics of the time, as the variousinterpreters of Plato’s final philosophy have shown. The philosophy over the theoryof ideal numbers and figures inspired Lautman15. We can, thus, definitively say that itwas this Platonic mathematics – or the mathematics of his time – that finally

15 See in particular [BEC 31], a journal edited by O. Neugebauer, J. Stenzel and O. Toeplitz.


influenced Lautman’s mathematical philosophy. This finally consisted of describing,as closely as possible to the mathematics that it produced, the speculativemorphologies that accompanied this.

11.3.3. Thom and the catastrophe theory

Lautman still remained within the field of purely abstract thought and themorphologies that translated this thought were conceptually described. In the 1970s,the French mathematician René Thom (winner of the 1958 Fields medal) tried, on thecontrary, to apply the theory of singularities of differentiable applications (of whichhe had been one of the promoters) to the qualitative study of physico-chemical,biological and linguistic phenomena. In short, he had opened up the path to amathematized philosophy of nature in itself, in its entirety. Founded on differentialtopology, but inspired by both Platonic and Aristotelian philosophy, Thom’s workalso, in a way, completed the Leibnizian project of a generalized analysis situs. Hehimself called this the theory of catastrophes, a name that must certainly havecontributed to the theory’s success.

Thom’s central project was to describe the world of phenomenological forms andthe discontinuities related to their emergence, their evolution and transformation, ortheir disappearance through qualitative models associated with an underlyingdynamic that is always presupposed, but which most often remains unknown. Theword “catastrophe” has no dramatic significance here. A “catastrophe” occurs when acontinuous variation of the underlying dynamic produces a discontinuous variation atthe level of phenomenological effects. This does, of course, concern the world ofappearances where, as Thom writes, “many familiar phenomena (to the point thatthey no longer attract attention) are ... difficult in theory” [THO 72, p. 10]. But, moregenerally, we can identify forms in all kinds of fields, concrete or abstract. In order todo this, it is enough to be able to associate them with a substrate space.

For Thom, as in the Gestalt theory, any phenomenological form stands out againsta background. In other words, it “expresses itself by a discontinuity with theproperties of the medium” [THO 72, p. 25]. This assumes that there is asupport-space or a substrate-space, as Thom sometimes calls it, which is most oftenan “open set” of space-time R4. Thus, as A. Boutot observes, “it is in this space thatthe forms of ordinary perception are described. But, in certain cases, the support forthe morphology may be much more complex” [BOU 93, p. 30]: functional space ofinfinite dimension describes the vibration of air in acoustics; spaces of moments orHilbert spaces in quantum physics [THO 74, p. 9], spaces of significant parameters insociology, etc.

Over the substrate space of the morphology, Thom isolated a box B containing thesystem being studied and observed the behavior of this system at every point (x, t) of


the product B!T of box B with the axis of time, T . The topology of the space of stateswill be accessible because of the measurement of the characteristic magnitudes thatcan be expressed in the functions g(x, t), which are not independent but are calculablefrom a small number among them. If we dispense with a precise quantitive modelwith numerical verifications, it is not necessary that these functions be analytical andit is enough to assume that they are m times continually differentiable. The internalspace of control for the dynamic will, thus, generally be a differential manifold M , asdescribed earlier.

The morphogenetic processes characterized by the formation, evolution ordisappearance of shapes within the box B are modeled by the data of a closed set Kin B ! T , such that at every point of K the process changes in appearance. A closedset, K, of this kind is called the set of catastrophic points of the process. It isassumed that it is not locally dense, to avoid the case of chaos or turbulence, wherethe problem of structural stability no longer arises. Elsewhere, that is, in the open setB ! T #K, which is the complement of the catastrophic point, the process is said tobe regular.

The distinction between catastrophic points and regular points is, clearly, relativeto the finesse of the means of observation used. It is therefore an idealization. But itallowed Thom to obtain a very general theory that could be applied to all morphologiesexperienced, from the most concrete forms of the everyday world to the categories oflanguage and thought, in general.

From the complex dynamic of the system X being studied, described by itsinternal space M , Thom would only retain the “attractors”16, as the rest of thedynamic only plays a “virtual” role, in his words. There is a “catastrophe” preciselywhen the attractor of the dynamic ceases to be structurally stable.

As the theory of dynamic systems and their attractors is reasonably complicated,Thom reduces it, initially, to the study of what we call “dynamics of gradient”, whichleads to a complete mathematical theory of “elementary catastrophes”. The vectorfield associated with a point x of the substrate is of the form X(x) = #

##+grad V ,

16 As Thom notes, it is not certain that a given field X in the manifold M always presentsattractors, a fortiori structurally stable attractors. Attractors may sometimes be infinite innumber (Newhouse) or present “an infinity of topological types for a dense sets of fieldsneighboring a given field (Lorentez attractors)” [THO 74, p. 41]. Nonetheless, according tocertain ideas postulated by S. Smale, if the manifold M is compact, almost any field wouldpresent a finite number of structurally stable attractors. Furthermore, on restricting ourselvesto isolated points and limited cycles (attractors of simple systems, but which are clearly only asmall part of the existing dynamic systems), the structural stability is immediate.


where V is a potential function over M (differentiable application defined over M inreal values) and

##+grad is the “gradient” operator that associates any real function with

a vector whose components are the first partial derivatives of the function.

Thom further hypothesizes that the manifold of internal states of the function isdifferentiable and compact, which implies that the attractors of the dynamic systemare uniquely isolated singular points. Indeed, over a compact manifold, the trajectoriesof gradient dynamic systems converge toward points of equilibrium, which are theminima of the potential function.

When the potential Vx, associated with the point in the substrate space x, has onlyone minimum c(x), the behavior of the system is well defined (e.g. a pendulum stopsat its point of equilibrium). But the potential function may present several minima.In this case, Thom applies two postulates that lead to two different definitions of thecatastrophic points and regular points.

The first is the Maxwell convention, according to which the system places itself inthe smallest of the minima. In this case, a point in the substrate is catastrophic in onlytwo cases:

1) When the associated potential Vx presents two absolutely equal minimas – thereis then conflict between the attractors that share, in the neighborhood of x, a domainof the substrate. We then have what Thom calls a “conflict catastrophe”.

2) When the absolute minimum of the potential, attained at a unique point m0,stops being stable – this is called a “bifurcation catastrophe” (in symmetry with“conflict catastrophe”).

The second postulate is the same as accepting that the process remains in thesame state, that is, in its minimum potential c(x), as long as it is not displaced fromhere in any way. If c(x) is stable, x is a regular point and the phenomenologicalappearance of the substrate varies continuously around the value c(u). The localprocess does not change in the phenomenological approach unless this potentialchanges topologically in the neighborhood of c(x), with the point x thus becoming a“catastrophic”’ point (bifurcation catastrophe). In this case, the minimum, c(x), isdestroyed, most often by collision with a local maximum, and ceases to be anattractor of the internal dynamic. At x, the substrate suddenly jumps from c to c1, thisbeing the new attractor corresponding to a more stable state (Vx(c1) < Vx(c)). Wesay that c1 has captured c. This is what happens in most applications.

The morphogenesis, thus, generally results from the emergence of an instability.The attractor of a dynamic, which has become unstable, the organizing center of thecatastrophe, “bifurcates” into two or more attractors that enter into conflict and giverise to a shockwave. These attractors themselves may bifurcate in turn and result innew conflict or bifurcation catastrophes.


The organizing potential of a catastrophe was called “logos” by Thom, afterHeraclitus. It corresponds, in fact, to a very precise mathematical function, which isnone other than the universal deployment of a singularity. A singularity, inmathematics, is a function that presents the particular feature of having all of its firstderivatives null at one point. A singularity is degenerate if all of its derivatives up to acertain order are null (at the same point). Thus, a function f(x) = xk is a degeneratesingularity of the order k # 1. Concentrating a global form into a point, this concealsa hidden complexity. The technique of universal deployment brings out this latentcomplexity which, in fact, hides the conflict of the local maxima and the minima. Thedeployment of the singularity, that is, its deformation under the effect of smallperturbations, brings out this latent structure. To obtain this result, we approach thefunction through its Taylor expansion and the universal deployment technique makesit possible to retain only the topologically pertinent deformations from thisexpansion. Thus, the universal deployment of the cubic singularity f(x) = x3 willhighlight a family of functions of one parameter Fu(x) = x3 + u(x). Generallyspeaking, one of Thom’s theorems is that any family of functions of m parametersmay arbitrarily be closely approached in Cs-topology by a family of m parameters offunctions where all of the singular points are isolated [THO 74, p. 75]. The numberof parameters q on which the universal deployment functions depend are called thedimension of universal deployment of f , or again, the co-dimension of the function.The co-rank of a singularity is the number of variables, k, that are pertinent from themorphogenetic point of view. The universal deployment thus operates a lamination ofthe function in strata in different co-dimensions, as the co-rank of singularity is thenthe minimum co-dimension of lamination that is best adapted to it.

In the case where k 6 2, these elements made it possible for Thom to give anexhaustive descriptive of the deployment of all singularities of co-dimension q ' 4.This is the theory of elementary catastrophes. These can be reduced to the followingclassification when we ignore the transversal intersection strata of the strata relative todisjointed singularities, which present no new morphology:

– Co-rank 1 catastrophes:

1) Strata with the co-dimension zero: V = x2 (simple minimum), thecorresponding points in the substrate space are regular points of the process. Itsuniversal deployment, V = x2, is none other than the function itself. Spatialinterpretation: being, object.

2) Strata with co-dimension 1: V = x3, universal deployment V = x3 + ux(stratum of the folded type). Spatial interpretation: edge, end.

3) Strata with co-dimension 2: V = x4, universal deploymentV = x4 + ux2 + vx (stratum of the wrinkled type or the Riemann–Hugoniotcatastrophe). Spatial interpretation: geological fault.


4) Strata with co-dimension 3: V = x5, universal deployment V = x5 + ux3 +vx2 + wx (stratum of the dovetail type). Spatial interpretation: slope, corner.

5) Strata of co-dimension 4: V = x6, universal deployment V = x6 + ux4 +vx3 + wx2 + tx (stratum of the butterfly type). Spatial interpretation: pouch, flake.

– Co-rank 2 catastrophes:

The classification here, founded on John Matter’s theory, takes into account thesignature of the quadratic form under consideration:

1) Mixed-signature quadratic form: V = x3 + y3, universal deploymentV = x3 + y3 + wxy # ux# vy (hyperbolic umbilic). Spatial interpretation: thecrest of a wave, arch.

2) Positive (or negative) defined quadratic form: V = x3 # 3xy2, universaldeployment V = x3 # 3xy2 + w(x2 + y2) # ux # uy (elliptical umbilic). Spatialinterpretation: needle, pike, hair.

3) Transition point between the previous two types: V = x2y + y4, universaldeployment V = x2y + y4 + wx2 + ty2 # ux # vy (parabolic umbilic). Spatialinterpretation: jet of water, mushroom, mouth.

Together, these seven elementary catastrophes allowed Thom to take into accounta large number of morphogeneses, both physical (in geology or hydrodynamics, forexample) and biological (notably in embryology).

The above-mentioned situation concerned pseudo-static forms, subject to agradient dynamics, the morphogenetic process being translated by an ordinarycatastrophe or being like a finite series of such catastrophes, as attractors areinterrelated through neighborhood relations or affiliation through the dynamic fiber.The final situation can be highly topologically complex without falling outside of thescope of a deterministic explanation. In the case of a metabolic form, which is verysensitive to disturbances, the recurrence of the dynamic fiber is destroyed and a newand brutal phenomenon intervenes, with the form dissolving almost instantly into acontinuum of elementary forms with simpler internal structure, static or metabolicforms of attractors that are smaller in dimension than the initial attractor c. Thomcalled this type of catabolic catastrophe, generalized catastrophes. A rough, purelyphenomenological classification shows the emergence of lump catastrophes, bubblecatastrophes, laminar or filamentous catastrophes or, again, catastrophes with spatialparameters. The overlaying of catastrophes can also be imagined, as in the case of themoiré effect, where two laminar catastrophes, with very regular periodicity, lead tothe formation of lump catastrophes ordered along alignments. Strictly speaking, thegeneralized catastrophes are not formalizable. Thom recognized this in the very firstedition of his book, but hoped that the evolution in work on qualitative dynamicswould make it possible to resolve the problem.


In summary, Thom’s general catastrophe theory leads to a global representationof all of nature based on the fundamentally deterministic models of differentialtopology. It opens onto a universal model that includes physical phenomena, life,social organization and even thought. A series of elementary or generalizedcatastrophes must, every time, be capable of describing morphogenetic processes.Although this was initially a purely local theory, the theory of catastrophes wouldthus arrive at a veritable cosmic synthesis, a philosophical mathematics of all ofnature.

It appears as though such a project had to be speedily abandoned, at least in itsmost ambitious form. As Thom recognized himself, his eminently respectable desireto “bring some order into the world” came up against mathematical impossibilitiesand irreducible physics such as nonlinear dynamics, systems sensitive to initialconditions and everything that the deterministic theory of chaos gradually helped usto understand about nature. Even within deterministic theories, Thom’s work wasthus limited by impassable obstacles. But the development of quantum physics andthe increasingly insistent presence of chance, including in mathematics ashighlighted by the algorithmic theory of information, challenged the pertinence ofthe mathematician’s conventionally deterministic vision. We can, of course, continuethinking, as David Ruelle does, that “any scientific study of the evolution of theuniverse must necessarily lead to a deterministic formula” [RUE 91, p. 42], but it isalso possible that such an assertion will remain, for a long time to come, if notforever, within the realm of wishful thinking. The theory of catastrophes has,moreover, helped us to demonstrate that mathematics is an indispensable guide forphilosophical thought and an inspiration that could lead to a renaissance in the fieldby offering a creative means of reviving old theories17.

17 Here, we can think in particular of the work carried out by R. Thom on Aristotle’s philosophy[THO 91], or again, the almost neo-Leibnizian projects that, in reality, constituted all of his workstarting from Stabilité structurelle et morphogénèse (Structural stability and morphogenesis).

12

Mathematical Research and Philosophy

In the following section, which acts as a guide to the analyses that mathematicianshave themselves carried on of their own research, we first review the different key-domains in mathematics in which remarkable advances have been made recently. Wethen study what philosophy has been able to take from these reflections or, rather,how these have been able to act as a source of inspiration for certain mathematicians1

allowing them to carry out a sui generis philosophical reflection.

12.1. The different domains

In a discipline as wide-ranging as contemporary mathematics, it is impossible tolist all the important sectors of activity and, a fortiori, to describe them exhaustively. Injust one century, the mathematical community of high-level researchers (grosso modo,those with a PhD in the field) has grown from a few hundred people to approximately80,000 people, according to Jean-Pierre Bourguignon, the president of the FrenchMathematical Society [BOU 02b]. Mathematical production around the world is, thus,vast and it has seen an exponential increase even within scientific production. Between1996 and 2011, an American researcher identified more than 25 million scientificstudies that were published, signed by approximately 15 million people around theworld. Mathematical output, among these studies, can be estimated at several milliondocuments.

The scale of the activities carried out by such large communities requiredclassification in order to not only make it easier to index publications andbibliographic research but also, at the institutional level, encourage orientation and

1 Because the professional philosophers and those who stay confined to their discipline wereknocked out of the field a long time ago.



the promotion of researchers within the institution. Established in conjunction withthe two main mathematical repertoires – the Mathematical Reviews (AMS) andSpringer’s Zentralblat MATH – it is now systematically used by bibliographicdepartments and is regularly updated based on the evolution of the mathematicalsciences.

It is organized by subject in a hierarchy of three levels. It includes multiplecross-references (proximity and thematic scope) and the taxonomy for the differentmathematical fields today comprises (if we restrict ourselves to mathematics in thestrictest sense) 65 main branches, 380 sections and more than 5,000 terminal leaves.The main branches are themselves grouped into five meta-branches (generalities andfoundations, discrete mathematics and algebra, analysis, geometry and topology,applied mathematics and others). The contents of this are given in Table 12.1, fromthe Mathematics Subject Classification (or MSC).

Generalities and Foundations00 General01 History and biography03 Mathematical logic and foundations04 This section was removed and set theory

is now part of the third branchDiscrete mathematics and algebra

05 Combinatorics06 Order, lattices, ordered algebraic structures08 General algebraic systems11 Number theory12 Field theory and polynomials13 Commutative algebra14 Algebraic geometry15 Linear and multilinear algebra; matrix theory16 Associative rings and algebras17 Nonassociative rings and algebras18 Category theory; homological algebra19 K-theory20 Group theory and generalizations22 Topological groups, Lie groups26 Real functions28 Measure and integration30 Functions of a complex variable31 Potential theory32 Several complex variables and analytic spaces

Mathematical Research and Philosophy 281

Analysis33 Special functions34 Ordinary differential equations35 Partial differential equations37 Dynamical systems and ergodic theory39 Difference and functional equations40 Sequences, series, summability41 Approximations and expansions42 Harmonic analysis on Euclidean spaces43 Abstract harmonic analysis44 Integral transforms, operational calculus45 Integral equations46 Functional analysis47 Operator theory49 Calculus of variations and optimal control; optimization

Geometry and Topology51 Geometry52 Convex and discrete geometry53 Differential geometry54 General topology55 Algebraic topology57 Manifolds and cell complexes58 Global analysis, analysis on manifolds

Applied Mathematics and Others60 Probability theory and stochastic processes62 Statistics65 Numerical analysis

Table 12.1. 2010 Mathematics Subject Classification (top levels)

Reduced to this list, such a set has all possible flaws:

1) It masks the multiple interconnections that may exist between the differentbranches of this “tree”, which are indicated notably by the multiple cross-referencesexisting at the level of the “leaves”. This signifies that the same subject could besituated in different places. For example, combinatorics is placed under the 5thbranch, but combinatorics for fields appears in the 11th branch (11Txx). Algebra andassociative rings appear in the 16th branch, but the commutative case appears in the13th branch. Lie groups are in the 22nd branch, but they also appear in the 54th


(54H15), 57th (57S) and 58th as transformation groups. “Particular functions” appearin branch 33, the branch that contains the properties of functions as functions, butorthogonal functions appear in the 42nd branch (42Cxx), combinatorial aspects arediscussed in 05 (05A), but aspects related to number theory are in branch 11, whilethose related to the theory of representation are in branch 22 (22E). There are anynumber of such examples.

2) A philosopher interested in comprehensive theories can, indeed, find a branchfor the “theory of categories” (branch 18) but a search here for “set theory” would befutile as it is, in fact, located in branch 33 (03E). This is undoubtedly proof of itsreduced importance or the diminishing of interest in this theory from the point ofview of mathematical research. Furthermore, Grothendieck’s K-theory finds amention in this classification, but not his topos theory or his theory of motives, whoseorganizational power is completely different.

3) Above all else, this nomenclature classification hides links that have beenestablished between domains that are as different in appearance as the number theoryand analysis, connected by the Langlands program [PAR 12b]. The subsequent workhas made these relations perfectly effective leading to the identification of similarobjects that will, undoubtedly, make a lasting impact on the way that mathematics istaught in coming years. It undoubtedly disregards, even more than did JeanDieudonné’s “panorama of pure mathematics” (which restricted itself to Bourbakimathematics), the existence of an organic architecture of mathematics, despite thefact that this constituted one of the objectives of the founders of Bourbaki [BOU 62].

12.2. The development of classical mathematics

We will not directly discuss the recent advances in the fields of probability oranalysis, which have now become very technical, but will focus on disciplines such asnumber theory, algebra and geometry, which seem to have had the greatest impact onphilosophy.

12.3. Number theory and algebra

Algebra and number theory have emerged from the traditional domains ofmathematics that, for over a century now, have had highly productive interactionswith other branches such as geometry or analysis.

Number theory has been highly successful these past few years, with the mostspectacular achievement being Andrew Wiles’ 1994 resolution of Fermat’s famousconjecture. This was presented by Fermat in the 17th Century as a “theorem” that heis likely to have demonstrated. In reality, the solution follows from the proof of a muchmore general recent conjecture, the Taniyama–Shimura–Weil conjecture, which made


use of practically every elaborate method in the arsenal of present-day mathematics.However, independent of this problem, there are still several important questions thatare open in number theory.

Thus, the comprehension of rational points of algebraic varieties defined over afield of numbers became the subject of many research projects carried out aroundthe Birch and Swinnerton-Dyer conjecture. From a rather arithmetic nature, the Artinconjecture on the functions, L, of fields of numbers endures even today. Let us alsomention Galois’ inverse problem (the conjecture that any finite group is a Galois groupof a field of numbers over Q), a problem that lies at the intersection of number theory,algebraic geometry and group theory.

Fermat’s problems gave rise to a whole lineage of mathematicians, includingHermite, Hadamard and then Weil, Serre and, above all, Grothendieck; this was as aresult of number theory appearing to be more and more closely linked with algebraicgeometry, from the time that we have been able to wield this discipline over finitefields, and not only in the now-conventional framework of complex numbers.

Interaction between arithmetic, geometric and algorithmic methods is, moreover,very strong and the point of origin of the work that breathed new life into questionssuch as that of continuous fractions had a well-known impact on the world of physics.We still await a new Plato, however, to revive the question of dichotomic processes ora linguist who could use these models in componential semantics.

In algebra, the most burning questions of the moment in finite and algebraicgroup theory are those that touch upon group theory and their representations thatconnect algebra and analysis. A certain number of conjectures, postulated in the last10 years, predicted strong links between combinatorics, groups, braid groups,algebraic geometry and topology and these have been corroborated by manynumerical results.

Similarly, several problems relating to the actions on reductive algebraic groupsremain open, such as the description of group actions on an affine space (withapplications to natural question in algebraic geometry) or the question of the generalproperties of quotients of open sets of projective algebraic varieties constructed usingthe theory of invariants.

Other conventional questions such as the calculation of multiplicities in tensorproducts of representations were newly clarified with the introduction of crystallinebases and work carried out on the model of paths. New formulae, purelycombinatorial, were obtained from here, which expressed the multiplicities as sumsof positive whole numbers. There has also recently been progress on thedecomposition of symmetric powers, but this question is one that is still beingworked on today.


Determining the irreducible characters of reductive groups as a positivecharacteristic (the focus of Lusztig’s conjecture) has been possible recently becauseof the bridging of the divide between quantum groups at one root of unity andKac–Moody algebras; it must be noted that one of these connections was establishedbased on ideas from theoretical physics.

Objects from convex geometry, polytopes and partition functions naturally appearin a number of problems that arise from the theory of reductive groups. These objectsalso play a role in current questions in symplectic geometry.

12.4. Geometry and algebraic topology

Geometry has developed into several branches (differential, algebraic, integral,etc.) that have their own problems and techniques. In particular, the new geometries,said to be “arithmetic geometries” (p-adic), were born and topology, the landmarkdiscipline of the 20th Century, also sparked off several focus areas.

Complex algebraic geometry, which was already greatly developed by the trioWeil–Serre–Grothendieck, was shaken up, very recently, by a dialogue withmathematical physics, especially field theory: we can, for example, cite the discoveryof mirror symmetry, a phenomenon of duality between families of the projectivemanifolds of a particular type, called “Calabi-Yau”.

Conjectures describe very precise relations between a manifold and its “mirror”,notably making it possible to enumerate rational curves over the manifold. Let usalso cite, among many others, the “non-abelian theta functions” over the space of themodules of principal bundles over curves, the Donaldson invariants and, most recently,the Seiberg–Witten invariables over algebraic surfaces.

Many of these theories now have a mathematical status, but not all: the intuition offield theory, the common factor driving all these discoveries, is presently beyond thescope of mathematicians. Making this a rigorous theory is a long-term objective but itmay be possible to develop a parallel mathematic intuition. Other, more conventionalfields remain very attractive. The Mori theory aims to describe the fine structure ofalgebraic varieties. It now fits in well in dimensions that are lower than or equal tothree, but does not fit well in higher dimensions. The methods used here are veryalgebraic; the case of non-algebraic complex varieties is very poorly understood.

Let us also cite the cosmology of the space of modules of curves, the difficultyin characterizing the jacobian of the curves among the abelian varieties (the Schottkyproblem), the search for simple criteria to decide whether a family of hypersurfacesover a variety may be divided by hyperplanes through appropriate embedding (theFujita conjecture), the theory of algebraic cycles (what type of subvarieties a givenvariety contains), etc.


Real algebraic geometry studies systems of polynomial equalities and inequalitiesin the field of the real. It emerged as a subdiscipline around 20 years ago. It drewtopologists as well as algebraists, geometricians and model theorists, while also beingthe source of many applications. The essential questions are posed around famoushistorical problems: the topology of real algebraic sets (Hilbert’s 16th problem), realalgebra and the sum of squares (Hilbert’s 17th problem), the relation between analyticfunctions and algebraic functions, semi-algebraic sets and generalizations, effectivityand algorithmic in the continuation of Sturm’s theorem, etc.

Theories of modeling topological spaces using algebraic objects have also beengreatly developed. After detailed knowledge of rational models that followed fromthe works of Quillen and Sullivan, several theories describing torsion were born.Finer, simplicial algebraic models made it possible to then take into account Steenrodoperations.

The homotopy theory, which essentially arose after Poincaré introduced thefundamental group of a space, now has as its central problem, the classification, up tocontinuous deformation, of continuous applications between spaces. Diversequestions motivated this research. Among them were the classification of spaces ofsmall dimension and the theory of manifolds via the Thom–Pontryagin construction.In the last 15 years, there have been considerable advances in the field of equivarianthomotopy, that is in the presence of the actions of groups, which led to the study ofgroup actions on manifolds. Major conjectures were resolved here. Moreover, thehomotopy theory resulted in strong links with algebraic k-theory and homologicalalgebra. Another spectacular development over time was the introduction of stablehomotopy techniques in a domain central to modern algebraic geometry, the theoryof schemas. The theory of homotopy has already witnessed several revolutions andalways offers vast fields for exploration.

Following Elie Cartan, Riemannian geometry is also alive and active, as are realanalysis, measure theory (following Lebesgue), functional analysis (thus named byPaul Lévy in 1924) and partial differential equations, sometimes presented as thestudy of inequalities and function spaces; their domain presents multiple interactionswith other fields of mathematics and the modeling of phenomena from disciplinessuch as physics, chemistry, biology, economics and imaging. Once these theoreticalfoundations are enriched by technical contributions that come from analysis,geometry and even algebra, new connections are formed with differential geometry,dynamic systems and probabilities. On the other hand, the domains of applicationsare only growing, from problems in chemistry such as combustion or kinematics ofreactions, to certain aspect of financial management.

The theory of dynamic systems, that of probabilities and statistics, in associationwith quantum phenomena and infinite particle systems have seen a large-scale


resurgence, especially as they are related to new fields that appeared recently instatistical physics and in graph theory, like percolation or the study of random media.

Furthermore, statisticians have created new tools to interpret random phenomena.These include wavelets, large deviations, geometry – models that are constantly usedin economics, medicine, industry and many other scientific disciplines.

Transversal themes such as quantum groups, automorphic functions (the field wasdeveloped more than 30 years ago from an imposing set of conjectures postulatedby the Canadian mathematician Robert Langlands and is the source of a potentialunification of geometry and algebra) or again, the theory of dynamic systems (far fromequilibrium thermodynamics, systems sensitive to initial conditions, foreign attractors,fractal structures) spread far and wide through mathematical culture.

Given the completely empirical diversity in the above-mentioned disciplines, itmay seem that in the project of transposing advances in the mathematical sciences tophilosophy, we have only fragments or tatters – in any case, elements that are veryscattered – rather than a body of established doctrine. Above all, the axiomatizationof theories (logic and mathematics) brings in multiple possibilities for thought thatcannot, thus, make up a single architecture as was done at the time of thePythagorean theory of medieties or at the time of the infinitesimal calculation or astopology emerged. Different world-views consequently appear to be juxtaposed but itdoes not seem possible to bring them together through any fact other than that oftheir shared language – mathematics. It is, thus, important on highlight the existenceof domains that lead to unification.

12.5. Category and sheaves: tools that help in globalization

Other than the language of sets and fundamental structures that can be constructedon them – algebraic structures, structures of order, topological structures (the motherstructures for “Bourbakism”) – mathematics of the 20th Century developed two formsof languages: that of the theory of categories and that of the theory of sheaves. Wewill elaborate on both these domains a little bit.

12.5.1. Category theory

The concepts of categories, functors, natural transformations, limits and co-limitsappeared in 1945 in a document by Eilenberg and Mac Lane titled, “General Theoryof Natural Equivalences”. An earlier study by these authors (1942) studied thesefunctors and natural transformations in the domain of groups. Their desire to clarifythese results led them to conceive of the category theory. To give meaning to theconcept of “natural transformation”, they introduced the term “functor”, borrowed


from Carnap, while the concept of category came from Aristotle’s philosophy viaKant and C. S. Peirce, but was defined here in mathematical terms.

For approximately 15 years, this theory was a useful language, especially forunderstanding algebraic topology or homological algebra and it was not certain thatthe role it played was greater than this.

The situation changed completely in 1957, however, with Grothendieck’shistorical article entitled “On Some Points of Homological Algebra”. The authorused categories to define and construct more general theories that he then applied tospecific domains such as algebraic geometry, for example.

A little later (1958), Kan showed that “adjoint functors” subsumed importantconcepts from the theory of categories, such as limits and co-limits and could,moreover, capture fundamental concepts from many other fields (in his case this wasthe theory of homotopy).

And then, notably through Lawvere’s influence (1966), the theory developed in afoundational sense, effectively expressing all forms of logic or deductive theories (see,in particular, Lambek’s work).

We will restrict ourselves to the essential – a few definitions and indicating howthis theory impacted philosophy.

DEFINITION.– A category C is the data from four elements:

1) a class whose elements are called “objects”;

2) a class whose elements are called “morphisms” (or “arrows”) and two“functions” (in the sense of “functional classes”), called source and target, from theclass of morphisms in that of objects; f : A + B signifies that f is a morphism “of Ain B” (i.e. with A as source and B as target) and the class of all these f morphisms isdenoted by Hom

,A,B

-;

3) a morphism idA : A + A, for every object A, called the identity over A;

4) a morphism g : f : A + C for every couple of f morphisms f : A + B andg : B + C, called the composite of f and g, such that the two following two axiomsare satisfied:

Associativity: for every morphism f : c + d, g : b + c and h : a + b, h =(f : g) : h = f : (g : h).

Identity: for every morphism f : A + B, idB : f = f = f : idA.

This definition, that we find in most textbooks, is in fact based on the language ofset theory. An alternative that Lawvere proposed in the mid-1960s consisted of


developing an intrinsic language founded on the concept of the categories of allcategories [LAW 65], a procedure that is still being debated.

Another solution, suggested by Lambek, and dating to this same period, is that ofviewing categories as deductive systems, objects as formulae, arrows as proofs andoperations on the arrows as rules of inference2.

As examples of concrete categories, we can cite:

1) The Set category, whose objects are sets and morphisms are usual functions.

2) The Top category whose objects are topological spaces and morphisms arecontinuous functions.

3) The Hotop category, whose objects are topological spaces and the morphismsare classes of equivalence of homotopic functions.

4) The Vec category, whose objects are vector spaces and the morphisms are linearapplications.

5) The Diff category, whose objects are differential varieties and the morphismsare smooth applications.

6) The DROP and poset categories, whose objects are pre-orders (respectively,partially ordered sets) and the morphisms are monotonic functions.

7) The Lat and Bool categories whose objects are lattices (respectively, Boolean,algebras) and the morphisms are homomorphisms that preserve these structures(through the operations: ;, <, 0, =).

8) The Heyt category whose objects are Heyting algebra and the morphisms arehomomorphisms that preserve this structure (through the operations: ;, <, 0,=, +).

9) The Mon category whose objects are monoids and the morphisms are monoidhomomorphisms.

10) The AbGrp category whose objects are abelian groups and the morphisms areabelian group homomorphisms.

11) The Grp category whose objects are groups and the morphisms are grouphomomorphisms.

12) The Ring category whose objects are rings (with unity) and the morphisms arering homomorphisms.

2 Let us mention here the three principal articles by the author on this subject [LAM 68, LAM69, LAM 72]. Let us also indicate, for philosophers, the interesting reflection by the same author[LAM 82].


13) The Field category whose objects are fields and the morphisms are fieldhomomorphisms.

14) Any deductive system T whose objects are formulae and the morphisms arethe proofs.

The theory of categories plays a unifying role in mathematics in two ways. First,any mathematical structure defined over a set leads to a category as long as it isassociated with the appropriate concept of homomorphism. Second, once a type ofstructure has been defined, it makes it possible to determine the conditions underwhich the other structures can be defined starting from this. For example, given twosets A and B, it is possible, in the category of sets (and in set theory), to form theCartesian product A!B.

Inversely, the category theory also makes it possible to say how the pre-definedtype of structure can itself be decomposed into substructures. Thus, given an abeliangroup, how can this be decomposed into more elementary structures? In all cases, itis necessary to know how structures of a certain type can be combined.

The category theory shows that many of these constructions are revealed to beobjects of a category possessing a “universal property” or that presents itself as thesolution to a “universal problem”. For example, a product of two objects, X and Y , ina category C, accompanied by two morphisms called “projections”, p : Z + X andq : Z + Y , can satisfy the following universal property: for all objects W , with themorphisms f : W + X and g : W + Y , there is a unique morphism h : W + Zsuch that p : h = f and q : h = g. We have thus defined one product of two objects,X and Y , not the product of these two objects. Products and objects are defined up toone (unique) isomorphism.

This proves that the nature of elements plays no role and only the relationsbetween these elements count, in the category theory. The category theory thusmakes it possible to compare these structures. For example, it indicates how, inalgebraic topology, the topological spaces are related to groups (and also to modules,rings, etc) in different ways (homology, co-homology, homotopy, K-theory, etc.).Morphisms between categories are given by functors, which map objects andmorphisms from one category into another such that the composition of themorphisms is preserved. In general, there are many functors between two givencategories, but the question of knowing how they are connected is suggested by thetheory itself, which thus gives rise to “natural transformations”.

These notions are important in themselves, but the heart of the category theory lieselsewhere. It is in the concept, introduced by Daniel Kahn in a 1956 article, publishedin 1958, on the “adjoint functor”, which is a sort of conceptual inverse.

Let us begin with an example. Let U : Grp + Set be the forgetful functorassociating any group with the set of its elements. The functor U is called this


because it “forgets” the structure of the group and the fact that the morphisms in thiscategory are group homomorphisms. The two categories are assuredly notisomorphic. The category of groups has a zero element, not the category of sets. Itwill certainly be impossible to find the inverse of the functor U , in the algebraic senseof the term. On the other hand, it is possible to construct a group from a set, basedsolely on the concept of the group and nothing else. We then say the constructedgroup is a “free” group.

There thus exists a functor F : Set + Grp that makes any set X correspondto the group F (X) over X , and every function f : X + Y , corresponding to thehomeomorphism F (f) : F (X) + F (Y ), defined in an obvious way. The fact thatU and F are conceptual inverses is formally expressed as follows: applying F andthen U does not give us X again, but there exists a fundamental relation / : X +UF (X) called adjunction unity, which simply associates each element of X withitself in UF (X) and which satisfies the following universal property: for any functiong : X + U(G), there is a unique group homomorphism h : F (X) + G such thatU(h) : / = g.

In other words, UF (X) is the best possible solution for the problem that consistsof making the elements of X the generators of a group (problem of insertion ofgenerators). By composing U and F in inverse order, we obtain a morphism. : FU(G) + G, called the co-unity of the adjunction, which satisfies the followinguniversal property: for any group homomorphism F (X) + G, there exists a uniquefunction h : X + U(G) such that . : F (h) = g : FU(G) constitutes the bestpossible solution to the problem of representation of G as the quotient of a freegroup. The identity (which, as we have said, is not algebraic) is then expressedthrough the following diagrams:

U&$U

!!

$$###

####

##UFU

U$&%%

U

FF$p

!!

$$$$$

$$$$

$$FUF

'$F%%

F

In the general case, the definition of an adjoint functor takes the following form:

DEFINITION.– Let F : C + D and G : D + C be functors going in oppositedirections. F is the left adjoint of G (G the right adjoint of F ), denoted by F > Gif natural transformations exist: / : IC + GF and . : FG + ID, such that thecomposites:

G&$G

!! GFGG$'

!! G


and:

FF$&

!! FGF'$F

!! F

are identity natural transformations.

Adjoint functors are unique up to an isomorphism. They are formally equivalentto the concept of “universal morphism” (or “universal construction”) and to that of the“representable functor”. The left and right adjoint functors preserve, respectively, theco-limits and limits of their domain.

Because of the concepts of this kind it has become possible to compare verydifferent mathematical structures and to think precisely of the relations that may existbetween them.

Thus, the forgetful functor U : AbGrp + AbMon of the category of abeliangroups in the category of abelian monoids admits a left-adjoint functor F : AbMon +AbGrp that, given an abelian monoid M , associates the best possible abelian group,F (M), with it such that M is a submonoid of F (M). For example, if M is the set ofnatural numbers N, F (N) is none other than Z.

Another example: there is a forgetful functor U : Haus + Top of the categoryof topological spaces that forgets the Hausdorff condition. Such a functor correspondsto the adjoint functor F : Top + Haus, such that F > U . Given a topological spaceX, F (X) associates the best Hausdorff space with it, constructed from X , namely thequotient of X through the closing of the diagonal !X ? X !X , which is a relationof equivalence.

Let us now consider the category of compact Hausdorff spaces, kHaus and thefunctor U : kHaus + Top, which forgets the compactness and the separation. Theleft adjoint of U is the Stone–Cech compactification.

We can offer diverse examples: all the fundamental operations of the theory ofcategories are born out of situations of adjunction of this type. And not only can wethus move from one structure to another, but we can also describe as adjoint alllogical theories that generally correspond to algebraic ordered structures (distributivelattices, Heyting algebra, Boolean algebra, etc.) and also the logical operatorsthemselves. Thus, the quantifiers are adjoint operations of substitution and, moregenerally, Lawvere was able to show that syntax and semantics are related throughadjoint functors [LAW 69].

The category theory, which poses many philosophical problems related to thenature of mathematical objects, logic and the question of the foundation ofmathematics, has resulted in few truly philosophical uses. This is undoubtedly


because of the abstraction of its formalism. We can only cite the proposition ofBaptiste Mélès [MEL 16], “The Classification of Philosophical Systems – fromEmmanuel Kant to Jules Vuillemin: An Architectonic, Logical and MathematicalStudy”. In the context of our own reflection on the idea of a general theory ofclassifications, we have also written a commentary on an article by R. S. Pierceentitled “Classification Problems”, which uses the theory of categories to define thecategories used for classification by rejecting all morphisms that are notisomorphisms, from all categories of mathematical structures. The comparison ofthese categories is assumed to give a general structure but does not always make itpossible to effectively construct the required classifications [PAR 13].

12.5.2. The Sheaf theory

The Sheaf theory, introduced by Jean Leray after the Second World War, byextending the work carried out during his period of captivity in Austria, was laterreformulated by Henri Cartan. His methods were then extended to algebraicgeometry by Jean-Pierre Serre, before being profoundly reworked and generalized byAlexander Grothendieck, and was finally used by Sato in the framework ofD-modules and micro-local analysis [HOU 98].

Initially, that is in 1946, Leray had envisaged a reconstruction of algebraictopology and sought to associate cohomological algebraic invariants with topologicalspaces. But Grothendieck saw, in this idea, a powerful tool that would make itpossible to ensure, in different branches of mathematics, a passage from the local tothe global. In order to fully leverage this, he invented the concept of a “presheaf”.

DEFINITION.– [presheaf] Let X be a topological space and C be a category. Apresheaf of objects F over X is given:

1) for any open set U ( X , of an object F(U) ( C, called “object of the sectionsof F over U (or above U )”;

2) for any open set V 7 U of a morphism #V U : F(U) + F(V ), called”morphism of restriction of U over V ”.

These elements are such that, for any inclusion of open sets W 7 V 7 U , we have:

#WU = #WV : #V U

F(X) is called “objects of global sections”.

In an equivalent manner, we can define a presheaf F : U @+F (U) as acontravariant functor of the category of open sets in X (with the inclusions asmorphisms) in C.


The most common presheaves take values in concrete categories (categories ofsets, groups, rings, vector spaces, algebra, modules, topological spaces, topologicalgroups, etc.). In this case, for any open set V 7 U , we write:

1s ( F(U), #V U (s) := s|V

and an element s ( F(U) is called a section of F above U . We write " (U,F) insteadof F(U).

A fundamental example of a presheaf is one where the restriction morphisms arethe usual restrictions of functions. Notably, over a differential manifold (respectively,an analytic manifold) X , for any open set U 7 X , the set C#(U,C) of functions thatare indefinitely derivable from U toward complexes (respectively, the set of analyticalfunctions with complex values O(U)) is a ring. These rings form a presheaf of ringsover X by considering the usual restrictions of the functions.

DEFINITION.– [sheaf] Take the example given earlier, of functions of the class C#

over a differential manifold X . The property of these functions of being indefinitelydifferentials is local. It is, thus, possible to “splice” the functions C# that coincideat the intersections of their domain of definition (including when this part is empty)at a global function C#. This would be the same for continuous functions or, moregenerally, functions of the class Cm. It is the same for distributions over aparacompact differential manifold of finite dimension, or for analytic functions orhyperfunctions over a real, paracompact analytic manifold of finite dimension.

A presheaf of sets F over X is called a presheaf when, for every open set V of X ,the union of a family of open sets {Vi}I , and for any family {si}I of sections of Fover the open set Vi, verifying:

si|Vi*Vj = sj |Vi*Vj

there is a unique section, s, of F over V such that:

s|Vi = si

Generally speaking, a presheaf F over X with values in a category C is called asheaf if the following condition is met: for any object T in C, U @+ HomC (T, F (U))is a sheaf of sets.

We thus define a sheaf of groups (respectively, abelian groups, rings, etc.) over atopological space x, as being a presheaf of basis X with values in the categories ofgroups (respectively, abelian, rings, etc.), which verifies the above condition.


There are sheaves of rings, modules, topological groups, etc. One of the interestingfeatures of this concept is that it gave rise to a generalization that led to the concept of“topos” in the sense that Grothendieck used it.

The sheaf F , as defined previously, is a functor of a particular type of categoryof open sets of a topological space in a category C. But a more general case can beconsidered: let S be a “small category” (e.g. a category where the class of objectsis a set) admitting product bundles and let C be a category. A presheaf F over Swith values in C is, in general, a contravariant functor of S toward C. S can thenbe equipped with a structure called “Grothendieck’s topology”. This is the same asdefining, for every object U in S , “covering families” of U , that is the families ofmorphisms Ui + U , which have properties analogous to the covering of an open setU in a topological space X by a family of open sets Ui 7 U , the morphisms in thiscase being the inclusions. The category S , equipped with a Grothendieck topology,is called a site. A sheaf on the site S with values in C is defined from the concept ofa presheaf by reasoning, mutatis mutandis, as if S was a usual topological space, anintersection of open parts being replaced by the product bundle.

12.5.3. Link to philosophy

Strictly speaking, there is no “application” of the mathematical theory of Sheaftheory in philosophy. Nevertheless, we can see that the Scottish philosopher DavidHume created a bundle theory, an ontological doctrine relative to objectness andaccording to which an object is made up of a collection (bundle) of properties,relations or tropes. According to this theory, an object is composed only of itsproperties: thus, there cannot be an object without properties and we cannot even“imagine” such an object. For example, the bundle theory claims that thinking of anapple necessarily means thinking of its color, its shape, the fact that it is a kind offruit, its cells, its taste or at least some other property it possesses. Thus, this theorycan affirm that the apple is nothing more than a collection of its properties. Inparticular, there is no substance in which the properties are inherent. Hume thus goesbeyond Locke’s empirical theory, which still preserved the concept of substance.

A small note: in mathematics, a bundle is not a sheaf, and the idea Hume has inmind is probably closer to the trivial concept of sheaf. Indeed, further reflection leadsto the view that Hume’s theory of thinking requires manipulating sets of continuousproperties that could be assembled into bundles or sheaves, which is not so far fromthe heart of the previous formalization.

DEFINITION.– [topos] Any category that is equivalent to the category of the sheavesof sets on a site is called topos.

The concept of topos generalizes that of the topological space. There are, however,many examples that have no relation to topology: if G is a group, the category of sets


over which G operates is a topos. The “punctual topos”, for example the category ofsheaves over a space reduced to a point, is none other than the category of sets.

12.5.4. Philosophical impact

The concept of “topos”, as we will see, would become the basis for the unifiedtheory of mathematics that Grothendieck tried to establish and that he then generalizedwith the concept of motive. Furthermore, the concept developed by Lawvere, of alogical topos, made it possible to reinterpret all logic in an entirely geometrical andtopological framework.

We wish to conclude with a brief presentation of Grothendieck’s unique vision ofmathematics, the organic nature of which seems, in itself, to provide manyphilosophical aspects.

12.6. Grothendieck’s unitary vision

As Pierre Cartier [CAR 00] explained, Grothendieck had laid out a fabulousresearch project, from 1958 or so, which had the goal of creating a veritable“arithmetic geometry”, going beyond the opposition of the discrete and thecontinuous by reworking algebraic geometry. On studying this in detail, we see thathe always pursued the greatest generality possible by appropriating the new toolscreated for topology, which had already been used by Cartan, Serre and Eilenberg.

12.6.1. Schemes

The theory was first organized around the concept of “scheme”, which was notonly the skeleton of an algebraic variety (as for Chevalley) but, as Cartier wrote,“was the focal point which was the source of all projections and all incarnations”.More precisely, Grothendieck went beyond two different points of view: that ofChevalley, for whom the concept of scheme of an algebraic variety was reduced tothe collection of local rings of the subvarieties; and that of Serre, who introducedalgebraic varieties based on Zariski’s topology and sheaves. Each approach presentedadvantages but also limitations: a field with a closed algebraic basis in Serre’sconcept, irreducible varieties in Chevalley’s concept. Chevalley’s point of viewseemed more appropriate for extending to arithmetic, as Nagata observed [NAG 56,KLE 05]. Grothendieck created a synthesis based essentially on the conceptualpresentations of Zariski, Chevalley and Nagata.

Schemes would then appear as a way to code systems of equations and thetransformations that could be carried out on them. For Grothendieck, a scheme was,in fact, an absolute object. If we call this scheme X , then according to a technique


(that is famous even today), the choice of a domain of constant corresponds to thechoice of another scheme S and a morphism "X : X #+ S.

In scheme theory, a commutative ring is identified with a scheme, its spectrum, buta homomorphism of the ring A toward the B corresponds, inversely, to a morphism ofthe spectrum of B toward A. Moreover, the spectrum of a field has a unique underlyingpoint (there are many different points in this sense). Consequently, the data of the fieldof definition, as included in the universal domain, correspond in reality to the data ofa scheme morphism "T of T in S. A solution from the “system of equations” X , withthe “domain of constants”, S, with values in the “universal domain” T , correspondsto a morphism, !, of T in X such that "T is the composite of ! and "X . In symbols,this is:

X

!X

%%

T

(&&%%%%%%%%

!T

!! S

[12.1]

This point of view had great simplicity and would also be highly productive. Itdid, however, imply a total paradigm shift. The scheme became the mechanism of theprocess that resulted in points in space. Diagram [12.1] corresponds to the fact that !is a T -point of the S-schema X , regardless of the S-schema T .

12.6.2. Topoi

Let us now come to the topos (plural: topoi) the definition of which was givenearlier. We have just seen that the geometry of schemes is a geometry with a largenumber of points, at least with this highly generalized concept of “’point” thatfollows from [12.1]. The topoi realize, on the other hand, a geometry without points.The idea of a geometry without points is a very old one. Euclidean conceptualizationdoes, indeed, have points, but we consider all geometric figures such as lines, planes,circles, etc. In the 17th Century, there was Cavalieri’s attempt to promote a geometryof indivisibles, but this was soon eclipsed by the infinitesimal calculus. Thus, it isonly in the modern age, following the success of the set theory, that we have got intothe habit of considering that any geometric figure is made up of a set of points. Formodernity, a line is the set of its points and is, therefore, not a primitive object but aderivative. Nevertheless, there is nothing to prevent proposing an axiomaticframework for geometry, where points, lines, planes, etc., are placed on an equalfooting. We thus know of the axiomatics of projective geometry (Birkhoff), wherethe primitive concept is that of flat, a concept that generalizes lines, planes, etc., in ann-dimensional plane and where the fundamental relation is that of incidence: the


point is on the line, the line is in the plane and so on. Mathematically, we consider aclass of (partially) ordered sets, called lattices, and a geometry corresponds to one ofthese lattices.

In the geometry of a topological space, especially in the use of sheaves, thelattices of the open parts occupy the principal place and the points are relativelysecondary. We can, thus, replace a topological space with the lattice of its open sets.This idea has, moreover, been proposed several times. But Grothendieck’s originalityresides in the fact that he went back to Riemann’s idea, according to whichmulti-value holomorphic functions exist in reality, not over the open sets of thecomplex plane, but over the spread-out Riemann surfaces. The spread-out Riemannsurfaces are projected one over the other and thus form the objects of a category.Moreover, a lattice is a particular case of a category, in which there is at most onetransformation between two given objects. Grothendieck thus proposed replacing thelattices of open sets by categories of open étalés sets. Adapted for algebraicgeometry, this idea resolves a fundamental difficulty related to the absence of atheorem of implicit functions for algebraic functions. Thus, he introduced the étalesite associated with a scheme. Sheaves may, thus, be considered as particular functorsover the lattice of open sets (itself seen as a category) and are thus generalized tospread-out étale sheaves that are specific functors on the étale site.

Grothendieck would play around with multiple variations on this theme, withgreat success in diverse problems of geometric constructions. The most significant ofthese successes was, undoubtedly, the possibility of defining homological (orcohomological)3 theory, which was necessary to attack Weil’s conjectures: what isnow called the l-adique étale cohomology of schemas. As P. Cartier [CAR 98] notes,this is the same as considering the category F(X) of all sheaves over X . Theconstructions over the topological spaces are then replaced by constructions over thecategory of sheaves. In an additional step of abstraction, an abstract concept of topos,the ultimate generalization of the concept of space, is discussed by Grothendieck,followed by Lawvere and then Tiernay. The category of sets thus becomes aparticular topos. Since the time of Cantor and Hilbert, work has essentially beencarried out in the category of sets. But, as Cartier writes, “Grothendieck claimed theright to retranscribe mathematics into any topos”4, thus founding the intuition of

3 “Cohomology” is a series of abelian groups defined based on a cochain complex (cochaincomplexes are algebraic tools that make it possible to study relations between cycles and edgesin the different dimensions of any space). Calculating a cohomology is, in general, showingthat two varieties are not homeomorphic or are not homotopic, depending on the type ofcohomologies studied.4 This mathematical theory of relativity is probably an additional reason for accepting thecomparison that Grothendieck carried out in Récoltes et Semailles, between his work and that


Brouwer and Heyting according to which the rules of intuitionist calculations aresimilar to the rules of manipulation of open sets: in any topos T, there is a logicalobject # whose elements are the truth values of the topos.. When T is the topos of theset, we have the conventional values (true/false) but when the topos T is that of thesheaves over a space X , the truth values correspond to the open sets of X .

12.6.3. Motives

Let us conclude with a few words on motives.

The reasons for the existence of the concept of “motive” come from theproliferation of cohomologies. It appeared toward the end of the 1950s that the Weilconjectures must follow from a purely algebraic cohomological theory possessing theright properties. Moreover, no cohomological theory with coefficients in Q waspossible and research was oriented toward a theory of a field of characteristic zero,different form Q.

In the early 1960s, Grothendieck proposed étale cohomologies and crystallinecohomologies and reformed De Rham’s cohomology in the algebraic framework,showing that it possessed the right properties at the characteristic zero. But there thenarose a jungle of “right” cohomological theories! If k is an algebraically closed fieldand 2 is a prime number different from the characteristic of k, the étale cohomologygives the 2-adic cohomology over Q), the De Rham cohomology gives groups thatare vector spaces over k, if the characteristic is non-zero, the crystalline cohomologygives the vector spaces over the field of fractions of the ring of Witt vectors withcoefficients in k, the Hodge cohomology makes it possible to associate certainvarieties with a Hodge structure, etc. The following situation thus arose: thesetheories could never coincide as they gave groups of cohomology over totallydifferent fields. However, they shared common properties that were grounds forthinking that they were from the same cohomological theory with coefficients in Q,even though such a theory does not exist. One possible approach is to initially restrictourselves to studying the case of the first cohomology group.

of Einstein. This comparison is added to those he mentions himself. “The comparison betweenmy contribution to the mathematics of my time and Einstein’s contribution to physics struckme for two reasons: both bodies of work were carried out to favor a mutation of the conceptionthat we had of ‘space’ (in the mathematical sense, in one case, and in the physical sense in theother); and both take the form of a unifying vision, embracing a vast multitude of phenomenaand situation which, hitherto, had appeared separate from one another. I see a clear mentalrelationship between his work and mine” [GRO 86, p. 68].


Grothendieck spoke of motives for the first time in a letter to Jean-Pierre Serre in1964. He used it to designate (following a musical analogy that would be repeatedlater in Récoltes et Semailles)5 the “common motive” (the “common reason”) behindthis multitude of cohomological invariants. The following is the text from Récoltes etSemailles:

“It was through my intention to give expression to this ‘kinship’ betweendiffering cohomological theories that I arrived at the notion of associating analgebraic variety with a ‘motive’. My intention in using this term is to suggest thenotion of the ‘common motive’ (or of the ‘common rationale’) subsidiary to the greatdiversity of cohomological invariants associated with the variety, owing to theenormous collection of cohomologies possible apriori. The differing cohomologicaltheories would then be merely so many differing thematic developments (each in the‘tempo’, the ‘key’, and ‘mode’ (‘major’ or ‘minor’) appropriate to it), of an identical‘basic motive’ (called the ‘motivic cohomological theory’), which would also be atthe same time the most fundamental, the ultimate ‘refinement’ of all the differingthematic incarnations (that is to say, of all the possible cohomological theories). Thusthe motive associated with an algebraic variety would constitute the ultimateinvariant, the invariant par excellence from the cohomological standpoint among somany musical ‘incarnations’, or differing ‘realizations’. All of the essential propertiesof the cohomology of the variety could already be read off (or be ‘extended to’) onthe corresponding motive, with the result that the properties and familiar structures ofparticular cohomological invariants, (l-adic, crystalline for example) would bemerely the faithful reflection of the properties and structures intrinsic to the motive.Here we find, expressed in the untechnical language of musical metaphor, thequintessence of an idea (both delicate and audacious at once), of virtually infantilesimplicity” [GRO 86].

Mazza, Voedovsky and Weibel seemed to wish to illustrate motives using themetaphor of a mountain: in one dimension, they go back to projecting a curve ofcompactified continuous level (an idealization of the path that must be followed to

5 The use of the concept of “motive” by the painter Paul Cézanne does not, therefore, seemto have played any role here. The Russian mathematician I. Manin [MAN 68] seems to be thesource of this legend, which was then repeated by C. Weibel in the record of V. Voevodsky,A. Suslin and E. M. Friedlander [WEI 02]. The legend was reinforced by the title page of thebook by Mazza, V. Voedovsky and Ch.Weibel [MAZ 06], representing the motivic formalisms(a morphism covering standard triplets) on the background of the Sainte Victoire mountain,while the back cover explained that “the concept of motive is an elusive concept, just likeits homonym, the ‘motif’ of the Impressionist method used by Cézanne in painting”. Misledby these allegations, a Chinese writer Xu Kejian went as far as to write an article to compareGrothendieck’s motives and Cézanne’s motives, though the two clearly have nothing in common[XU 12].


ascend?) over a compact series of discrete markers. In the cover image of their book,Lecture Notes on Motivic Cohomology, morphism is the relative dimension 1 and Zand X# are subschemas of X . It is enough for certain conditions to be satisfied to beable to define a covering morphism f of Y in X: in this case,f(Y ) 7 X, f |Y : X + Y is spread-out and there is an isomorphism: ZY + ZX ,where Zy = f&1ZX 3 Y [MAZ 06, pp. 84, 175].

Grothendieck, would never publish any explicit writing on the subject of“motives”. The letters sent to Serra are private correspondence and his polycopyentitled Motives was not, initially, meant for distribution6. It was MichelDemazure and Steven Kleiman who wrote the first articles based on his lessons. Whatthe mathematician had worked toward was, in fact, a universal cohomological theorythat would be based on the following structure: we consider a category of projectivevarieties and then:

1) the morphisms are replaced by the classes of equivalence of Q-correspondences ;

2) we formally add objects (kernels and projector images) to render the abeliancategory and be able to write a Künneth formula.

By using C to denote the category thus obtained and H to denote the dual categoryof C, the natural contravariant functor of the category of smooth algebraic varietiesin H can be factorized (by construction) through any cohomological theory of ourchoice: this is the desired theory of motives and the corresponding category is called“the category of motives”.

The chief defect of this construction is that it is not explicit. Worse: many difficultconjectures come in before being able to talk of correspondences or, more simply,algebraic cycles: Hodge conjectures, Tate conjectures, etc. These hypotheses todayform what are called the standard conjectures and it appears that any theory ofmotives, however partial and conjectural it is, depends on them. We thus only havethe fragments of such a theory, despite the work carried out by Manin, Chow,Bellison, Deligne and Voevodsky. We thus fall very short of Grothendieck’sexpectations.

According to Luc Illusie, Grothendieck “had a vision of global harmony inmathematics and intuition showed him the simplest path. His goal was to find the’universal ferment’, the profound unity of mathematics, by rising by degrees”. Themotives, his final creation, were thus a sort of all-encompassing meta-theory, themost vast that he could imagine. This was almost immediately confined to

6 The opus that bears this title is accompanied by the remark “do not disseminate”.


metaphysics. “How can we not sense behind the ’motives’ the idea of God?” Illusieconfessed, in another interview in the mid-1990 [ILL 95]. What Grothendieck saidmore or less confirms this idea.

12.6.4. Philosophical consequences of motives

It is not surprising to see Grothendieck’s evolution toward metaphysics and even acertain mystical spirituality.

In Récoltes et Semailles, he only laid out an organic vision of mathematics, withwhich a certain innocence is assumed to be associated. In a completely disinterestedregister, he deplores the recent evolution of mathematics that has abandoned thisinnocence as well as a certain ethics of truth and respect for the masters.

However, in later more “meditative” texts such as La clef des songes or Notespour la clef des songes, a spiritual quest becomes increasingly clear and the internaldialogue with that which Grothendieck called “the Dreamer”, because it is that whichdreams in us and carries our more fertile projects, develops without limits.

As we recalled earlier, the aim of Grothendieck’s mathematical creation wasalways to construct a set of more and more general structures that made it possible,finally, to reinterpret the whole domain of mathematics and, in increasingly greaterdetail, the universe itself.

One of the characteristic stylistic traits of Grothendieck is that he related vastcategories through morphisms articulated in diagrams. The diagrams associatecertain objects or structures with one another, using arrows that must satisfy certainconditions. We say, in particular, that a diagram is commutative if, when we choosetwo objects, we can follow any path through the diagram and obtain the same resultby the composition of the morphisms.

Commutative diagrams abound in Grothendieck’s work. This goes from the classiccomplex homomorphism of A-modules of SGA 2 (1962) [GRO 68], such that:

X' x!!

u

%%

CX'Cu

%%

Y'y

!! CY'

to the diagram of motivic categories in an unpublished article on motives, whicharticulated the Z-module M+(X) admitting as base the class of simple objects of thecategory of effective motives M+(X) and the set of its classes of simple objects


%+(X), which leads to a filtration over the ring M+(X) whose analog can be foundover M(X) with %(X) and whose graduation can then be expressed on M+(X):

M+(X)+

!!

*%%

Z"+(X)

*%%

M(X)+

!! Z"+(X)

Jean-Pierre Serre was, on one occasion, very moved by this proliferation. As hewrote in a letter to Grothendieck, about the SGA 5 (a work regarding which Illusiestated that he had not been able to verify the compatibility of all the diagrams): “wecannot stop with saying that the diagrams that we have written ‘must’ commute,especially when things as serious (for me ...) as the Weil conjecture or the Ramanujanconjecture depend on it! (If the natural ‘algebraic duality-arithmetic duality’ diagramcommuted, we could deduce from this that " is algebraic, as you know, and even that2i" = 1)” [GRO 04].

It can, of course, be guessed that in the domain of motives, where correspondencesare still rather conjectural, this observation can be applied more fully. But followinghis logic to its natural end of comprehensiveness and generalization, Grothendieckgave himself up to a meditation where mathematics no longer played any role but thatof an inductor, approaching the theological domain of the more nebulous ideas thatIllusie spoke about and that, to the mathematician, were undoubtedly hidden behindthe category of all motives.

In mystical writing, or when caught in periodic crises, the mathematician believedthat he had to battle with that which Thomas Mann called “the complete other” butwhich he simply called the “Devil”, without being able to remove it from hisglobalizing reflection. He thus laid the path for a sort of theosophy and – perhapsrecalling his time as a mathematician? – summarized this in a diagram7 articulatedaround different instances (the Mother, the Father, Lucifera, Lucifer, the good God,the shadow-God, yin-God and the yang-God) and forming the extravagant figure of aGod in nine people, which he presented as follows:

7 See A. Grothendieck, Développements sur la Lettre de la Bonne Nouvelle, unpublishedmanuscript, 1990.


Mothergood-God

!!

yin-God%%

Father[d]yang-GodGod

Lucifera[r]shadow-God Lucifer

This is, of course, complete mystical folly. But Grothendieck seems to have laterdisowned this text, which was, nonetheless, sent to 250 people he believed, at the time,to be the Chosen, destined for a grand mission. The “Correction”, addressed to thesesame correspondents, 3 months later, indicated that he was no longer certain of thetruth of the revelations described in La Lettre de la Bonne Nouvelle (The Letter of theGood News) and notably confessed:

“I was the victim of a mystification by one of the most [wicked?] ‘spirits’ (amongwhom I could not distinguish with my limited abilities) who was, I have not theslightest doubt, invested with prodigious powers over my body and my psyche”[JAC 04].

Whatever this explanation (which itself sounds quite unhinged) may mean, itcannot be contested that this second, very strange part of Grothendieck’s work didhave, as Cartier, Illusie and Maltsiniotis thought, an undeniable link with the firstpart, whether this was a perverse effect or a point of culmination, related to themathematician’s history and his personality, the same aspect that was able to producehis most beautiful theorems and his most elegant methods.

Whether we wish it or not, Grothendieck’s psychology, and also the mathematicsthat this produced, in his generalizing perspective, were both at the origin of his finaltheosophical mania and, whatever we may think of this evolution, we must accept it.

Conclusion

For a Philosophical Mathematics

We have finally come to the conclusion of the long reflection that we carried outon the relations between mathematics and philosophy. In the course of this reflection,we have accorded much importance to the first of these disciplines, which, to us,often inspires the other, gives it organizational schemes and both guides and restrainsthe imagination of the constructors of systems and the inventiveness that couldsometimes lead them to stray too far1. The reader could, therefore, legitimately ask:if the real driving force behind the evolution of thought is to be found within thesciences, especially mathematics, in the framework of operational language and notin its speculative extrapolations (which are often approximative and sometimeshazardous), then what is the meaning of these philosophical constructions, which donot have the goal of being operational nor of constructing objects, strictly speaking;which exhaust themselves in projects that are grandiose but ineffective and mostoften abortive; and whose operative character cannot have escaped anyone? In otherwords, and more clearly, why philosophy?

We know how Kant answered this question: besides the understanding, whosecognitive function is to associate concepts and intuitions (“pure” or “empirical”),there exists what Kant calls Reason, whose ideas, falling beyond the scope of theexperiential, amount to an unconditioned that contains the synthetic principle of theconditioned. These could not, however, provide the substance of knowledge as,according to Kant, they only played the role of “regulator”. Thus, in addition to “thepossibility of explaining” (which is the possibility of understanding confronted withan experience that is always open and of an essentially processual nature), we alsohave the necessity of understanding (in the sense of the German verstehen or

1 Malebranche, even in his time, had already castigated “the inventors of new systems and theimagination that characterizes them” [MAL 46].



bregreifen, or again from the Latin intelligere), that is, reporting what is at amovement of totalization, which goes beyond the visible and attempts to seize uponthe connection between each thing in order to best situate it within the All.

In theory, of course, humans could abstain from such a reflection, contentingthemselves with the simple life like other animals or, as the Empiricists sometimesadvise, renouncing such a process a priori because, given that everything is patentlyuntrue, we are wasting our time in trying to extrapolate it. But a certain worrypersists, demanding an answer to the enigma that is our world. Faced with theposition that consists of keeping silent or, as Wittgenstein recommended, every timea theorist fails to assign meaning to certain signs that they use, point this out to themand return to the discourse of science alone, there is perhaps place for anothermethod. Wagering on the existence of a truth in the world and, correlatively,wagering on the existence of a word that is capable of liberating a phenomenon fromits appearance, this method would strive to reveal the entirety of the hiddenarchitecture.

Comparing speculative language to scientific language, Jean Ladrière, in aremarkable article that has already been cited, sought to develop this theme, whichwent beyond the Kantian response. Contrary to what Kant thought, the propositionsof a scientific theory do not already have the goal of “categorically specifying, eachin its own way, a certain intuitive fact, but rather of establishing a system of relationsthat structure a certain semantic field” [LAD 71, p. 270]. Thus, scientific language(which is more independent of experience than the Königsberg philosopher thought)already has “a global correspondence, that is, moreover, always hypothetical andpresumptive, between the theory taken as whole and the set of experimental results”[LAD 71, p. 270]. These are, moreover, expressed in a language that has all of theproperties of a theoretical language, leading to a systematization of the objects of theexperience that is in no way the perceived world.

In this context, it then seems as if it is possible “to restore to the discourse ofreason a speculative charge much greater than Kant believed could be credited to it”[LAD 71, p. 270]. There would, thus, be the goal of totality that could be expressedrationally and a discourse whose purpose would be to speak of the world by talking ofthe unconditioned without fear of being reduced to a flatus vocis.

The semantic functioning of such a discourse would be a sort of transformationalfilter. It would consist of making a floating vision explicit, of moving from anorganizational scheme, which is only hinted at by a partial system, to a conceptualarticulation that is adapted to the initial vision:

“The art of the philosopher (Ladrière writes) ... consists of finding newphrases that guarantee propitious encounters, of arranging theirdiscourse so as to definitively bring out a novel, organon of meanings,

Conclusion 307

capable of substituting the incertitude of a vision with the objective rigorof an utterance” [LAR 71, p. 280].

On this point, thus, there is a strong relation between philosophical and scientificdiscourse. But while the scientific method is laid out within the framework of theoperational “that recommends and justifies use of mathematical language” [LAR 71,p. 280], philosophical discourse, without modeling itself on the structure ofexperience, would have to, in order to act, dig deep within its own resources, whichwere never been completely devalued, as it could always be “reworked in the life oflanguage, charged anew with significance and reanimated in an appropriate context”[LAR 71, p. 280] Thus, the meanings entered into the available semantic fields, aswell as those that still float around in the indeterminate state of the unsaid, could beendlessly re-articulated.

It is through such a discourse that the meeting of the logos and the universe iscarried out each time and through this we can “finally understand the secret of theircorrespondence, the reasons that make our speech effective and make the worldcomprehensible to us” [LAR 71, p. 281].

However, in mobilizing the dormant meanings in order to rise above them, such acourse would then be confronted with an infinite task that could only lead to a partialrealization: condemned to endlessly pursue the task of nomination, relating terms andpropositions, such a discourse would be condemned to inscribe the infinite networkof this in a system. Moreover, the distance between this system and the horizon isinfinite. Hence, the idea that any speculative effort would remain hazardous byessence, unstable and only premonitory. Also, thus this inevitable consequence thatWhitehead stated, whereby “every philosophy will one day be dethroned”.

The fact remains that in order to stay in the neighborhood of the real world, andavoid escaping into a visionary dream (or, as Kant said, into some dubious“Schwarmerei”), speculative discourse should base itself on the guidelines offered bythe scientific structuration of experience. This is, moreover, what most of the “great”philosophers have done.

In a book that is now already old [PAR 93b], we elaborated on these links betweenphilosophy and science in the following manner. The history of philosophy teachesus that any philosophical thought of any scale, any doctrine or “system” in the usualsense of the term, associates an argumentative apparatus (let us call it “logic”, forbrevity’s sake) and a general organizational structure (an “architectonic”) that MartialGueroult, making use of a musical metaphor, once compared to a “fundamental note”,which, even if it is never struck, would continue to exist in the multiple variationsof the melodic line: a sort of “motif” as Grothendieck would perhaps have said. Wethought that the origin of such a structure, when we attempted to locate it, could be


found in mathematics, generally in the most advanced or most comprehensive aspectof mathematics at that time.

We thus assume that at a given time t, a philosophical doctrine S# is related to amathematical theory TM , which can be designated by an arrow of T in S, to indicatethe possible existence of a partial homology between certain statements of T andcertain statements of S. We then imagine that at a later time, the theory TM has beenreinterpreted as the theory T !

M , which is usually more general and more powerful thanTM . Correlatively, at the same time, the interpretation I(S#) of S#, if we follow thecommon principle of mathematical generalization, must correspond to T !

M such that,as we say in homological algebra, the diagram is “commutative”.

TM

x!!

u

%%

S#

v

%%

T !M

y!! I(S#)

We have verified correspondences of this kind throughout this book and there isno need to review them here.

However, two problems could arise:

1) The first problem consists of knowing whether these homologies could continueto exist in a situation where mathematics has become greatly complexified. Giventhe complexification of the discipline, the project of establishing a philosophy thatcorresponds with the mathematics of its time could come up against a bewilderingnumber of potential inductor elements. Multiple possibilities of thinking arose as aresult of the axiomatization of theories in the 20th Century, and not all of them can fitinto the same architecture, as we have already said. Different views of the world can,consequently, be juxtaposed today without necessarily being brought together on anygrounds other than that of their shared language of mathematics.

2) The philosophical counterpoint to this situation is, perhaps, the repugnancethat philosophers encounter today in developing their thinking into a coherentsystem. This is not a new compunction. Not only has non-systematic thought existedsince antiquity and been maintained among aphoristic thinkers up to Cioran, but“suspicions” vis-à-vis the system (its “lack of probity”, Nietzsche said) have beenmade manifest in an explicit manner since the 19th Century by authors such asKierkegaard and have resulted, in the 20th Century, in a rejection of the idea of

Conclusion 309

“totality”, favoring instead that of “infinity”, especially in the work of Rosenzweigand Levinas.

In other words, we can conclude that philosophers today neither can nor want toconstruct “systems”, which, ipso facto, will render obsolete the “translations” that wehave mentioned, which would thus only refer to the past.

In an even older book [PAR 06], we defined the conventional philosophicalsystematicity using what we called the “Gueroult conditions”. As far as this historianand philosopher was concerned, every system assumed: (1) a representation of all ofreality; (2) maximum synthesis of all possible determinations; (3) the realization of acertain identity between the interior of the system and the external world, and finally,(4) the existence of a unique and absolute response to the question of the world andthe human in the world. We added, at the time, a fifth condition to this: the “Ladrièreprinciple”, which desires that any system, in the conventional sense of the term, be aclosed set of speculative categories whose course necessarily leads us back to thestarting point (the equivalent of a Hamiltonian path in a graph). We did not, however,exclude the possibility of developing “a generalized non-gueroultism”. It may, westate, take the form of methodological indifference, for example, toward the idea ofthe system or – correlatively – that of a multiplicity of accords without constraint thatcould include a certain randomness.

The study of Grothendieck’s final texts, and notably the topos theory and theoryof motives, perhaps allows us to glimpse at a new result today, one that may alreadyhave been anticipated by Michel Serres.

Confronted by the number of mathematical “models” of the Leibnizian system,Serres seemed to suggest that a system of this kind could, in some way, beretranslated or reinterpreted in any of these “models”. Combinatorial and reticulated,the Leibnizian system thought of rigor in terms of multilinearity and multivalence.Serres wrote, “these two characteristics are visible in two ways: through anintegration of a given concept with different orders; and by the fact that a givendiscourse, which can be taken to analyze a singular problem, along the way alsoanalyzes some others in an analogous manner. Such a text leaves open the possibilityof its translation across levels that are different from the level it is taken to be locatedon” [SER 68, vol. I, p. 16]. In doing this, Leibniz avoided unique chaining bymultiplying it. In this sense, Serres wrote again, “philosophy tends to becomemathematical, but the horizon of this future is a mathematics that is inconceivable inits own time” [SER 68, vol. I, p. 18].

It may be that in this description, Serres saw much further than Leibniz himself:he recognized, moreover, that the author “rarely says that his system forms such anetwork” [SER 68, vol. I, p. 28] and that it was therefore essential “to define, in hiswork, a certain distance between the hope and the realization, between the project and


its effectuation”. However, it is also possible that Serres saw much further than evenhe knew.

This is because, whether or not this conforms to Leibniz historically, hisdescription of this multilinear and multivalent systematicity coincides perfectly withwhat Grothendieck was doing in mathematics in the 1960s, when Serres was writinghis text. As we have seen, Grothendieck’s mathematics, especially via the concept oftopos as Cartier interpreted it, is indeed an attempt to rewrite the whole ofmathematics into any topos , just as Leibniz’s philosophy, as Serres saw it, at any rateretranslated the entirety of his thought into any region of the system, privileging noparticular path. If any neo-Leibnizian philosopher emerges today, there is no doubtthat it is in Grothendieck’s mathematics that they would find the ultimate result ofwhat Leibniz wished to do, and the meta-model for today’s philosophy itself.

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Index

A

absolute, 121, 137, 138, 142, 218, 248Archimedes, 4, 30, 51, 53, 55, 56, 59, 61,

63, 71, 73, 77, 84, 99, 120, 179, 199architecture, 286

conceptual, 306Greek, 11mathematical, 308

Archytas, 8Archytas’ curve, 34, 35Arcy (d’) Thomson, 16Aristotle, 18, 53, 61, 65, 66, 92, 96–98,

179, 180, 183, 200, 203, 278, 287

B

Bachelard, 21, 48, 145, 146, 187–189, 194,195, 200, 262

Baire, 214–216Banach, 139, 264–266Berger, 249Bergson, 150–152, 201, 262Beth, 221Birkhoff, 235Bolzano, 204, 205Boole, 234, 235, 237, 288, 291Borel, 165, 169, 175, 214–216Bourbaki, 76, 213, 281Bourguignon, 279Boutot, 273Broglie, 152

Brunschvicg, 9, 143, 146, 167, 186, 192,195

Bruter, 150

C

Cantor, G., 67, 91, 147, 203, 205–210,212–214, 217–220, 297

Cantor, M., 59Carnap, 75, 223, 271, 287Carnot, 74, 118Carrega, 67Carse, 219Cartier, 295, 297, 303, 309Cavaillès, 205, 218Cayley, 139, 196, 197, 200, 231, 247, 248Cézanne, 299Clifford, 242, 245–250coincidence of extremities, 59, 61Comte, 143, 144, 147, 148Condorcet, 172conics, 23, 30, 39, 71, 77, 79–81, 179, 239,

248continuous fractions, 13, 14, 65, 66, 94Cournot, 173Couturat, 209Coxeter, 184Crépel, 172, 173Cusa (of), 4, 59, 61–63, 68



D

Dahan, 99Dauben, 203Davis, 138Dedekind, 48, 205, 213, 216, 220, 238Delahaye, 176Descartes, 30, 39, 63, 73–79, 81–83,

85–89, 98, 101, 107–109, 122, 123,204, 225, 233, 248

Devereux, 48dichotomic processes, 16, 66, 219, 283Dieudonné, 76, 232, 281Diogenes Laërtius, 9disorder, 12, 110, 113, 117Duhem, 152

E, F

Eternel return, 11Faraday, 152Fedi, 147Finsler, 200, 201, 264Fourier, 145, 146, 172Fréchet, 254Frege, 222, 225

G

Gergonne, 213Ghyka, 8, 15Gödel, 212, 216–218, 224–226Graham, 152Granger, 66, 78, 89, 242Grassmann, 231, 241, 245, 249, 251Gregory, 64, 101, 113Grothendieck, 232, 280, 283, 284, 287,

292, 294, 295, 297–303, 309, 310

H

Hàjec, 172Hausdorff, 212, 214, 217, 256, 257, 264,

291Heath, 13Hegel, 74, 121, 131, 142, 204Heidegger, 218, 262Herzberg, 171, 172Hoëné-Wronski, 122, 137Houzel, 116

Hume, 294Husserl, 75, 219–221, 225, 226

I, J

ideals, 18Illusie, 300, 302, 303irrational, 4, 8–11, 13, 14, 16, 21, 51,

65–67, 92–94, 210, 212, 216Jackson, 303Jullien, 75

K

Kalmar, 203Kant, 201, 287, 292, 305–307Kierkegaard, 308Kleiman, 300Klein, 191, 196–198, 239Kucharski, 94Kurepa, 221

L

Ladrière, 225, 269, 306, 309Lagrange, 76, 113, 114, 117–119, 137,

139, 142, 144, 145, 220, 238Lambert, 65, 66Lascoux, 137Lautman, 270–272Le Lionnais, 63Lebesgue, 214–216, 285Leibniz, 63, 64, 74, 77–79, 84, 89, 99,

101–117, 147, 175, 181, 204, 225, 237,241, 253, 309, 310

Levinas, 308Lindemann, 47, 67, 68Locke, 105Loève, 162Lusin, 215, 216, 221

M

Malebranche, 305Manin, 299Mathesis, 76, 88, 219, 225Mazza, 299McClennen, 171medieties, 8, 27, 36, 93, 286Merleau-Ponty, 64, 201

Index 329

Mersenne, 84Michel, 13motives, 298–302, 309Mugler, 11Muir, 139multiplicities, 219, 220, 283Musil, 172, 174

N, O

Nagata, 295Nelson, 171Neumann (von), 170Newton, 74, 76, 99–101, 103–105, 113,

117–119Nicomachus of Gerasa, 8Nietzsche, 174order, 53, 76, 78, 88, 89, 93, 94, 104,

110–118, 141, 142, 146, 152, 162, 199,205, 209, 210, 212, 217, 221, 237, 248,263, 276, 278, 286, 290, 309Apollinian, 11and measurement, 78measurable, 93

Ostwald, 150, 151

P, Q

Pacioli, 15, 73Pappus, 8, 71, 81Parrochia, 174Perrin, 170Plato, 3, 9–12, 16–18, 20, 27–31, 35, 36,

40, 66, 93, 94, 179, 183, 231, 283Poisson, 172Prat, 146, 147Proclus, 8Proportion, 11, 14, 28, 29, 81Pythagoras, 7Quételet, 173Quine, 222, 223

R

Ramanujan, 16rationality, 10, 21, 24, 66, 75, 147, 188Renou, 92, 93Renouvier, 146, 147Rey, 54

Riemann, 73, 119, 176, 183, 184, 191, 192,194, 196, 197, 200, 213, 219, 254, 263,276, 297

Robinson, 120, 171Rosenzweig, 308Ruelle, 278Russell, 75, 213, 222, 223, 225, 271

S

Saint Augustin, 206Sartre, 219Schelling, 142schemas, 240, 285, 295–297Schilpp, 203Schopenhauer, 12series

algorithmically incompressible,174

Fibonacci, 15groups, 297infinite, 14, 65, 94, 98of numbers, 56, 102, 113, 116, 117,

174, 204, 208, 209of points, 83, 115, 256random, 174stationary, 165subdivisions, 114

Serre, 283, 284, 292, 295, 299, 302Serres, 146Sheaf, 286, 292–295, 297, 298Sierpinski, 216Simplicius, 53Sinaceur, 238Spengler, 12Spinoza, 118, 204, 235, 237, 248,

263square, 4, 23, 49, 51–55, 59, 61, 63, 99,

102Stewart, 233Stobaeus, 12Suslin, 215, 216system, 16, 34, 36, 71, 72, 78, 82, 86, 88,

93, 112, 115, 123, 137, 142, 151, 225,175, 176, 181, 185, 191, 216, 219, 233,235, 237, 247, 254, 266, 269, 270,273–275, 278, 285, 286, 289, 292, 296,306–309


T

Theon of Smyrna, 8, 14theory, 219

of a set, 10, 75, 164, 205, 207, 213,214, 216–219, 221, 226, 280, 281,287, 289, 296

of categories, 75, 280, 286, 287, 289,291, 292

of categories of Eilenberg-MacLane,286

Thom, 262, 273–278, 285Thomson, 152topological space, 214, 215, 255–258,

291–294, 297topos, 294–298, 309transcendance, 51, 68

V, W

Vitrac, 43Voedovsky, 299Vuillemin, 14, 20, 86–89, 292Waerden (van der), 246Wang, 225Weibel, 299Whitehead, 269, 270, 307Wittgenstein, 222, 223, 271, 306

X, Z

Xu, 299Zeising, 15Zermelo, 212–214, 217

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